HYDROGEN BOND, PI-PI STACKING, AND VAN DER WAALS INTERACTIONS

INVESTIGATED WITH DENSITY FUNCTIONAL THEORY

AN ABSTRACT

SUBMITED ON THE THIRTIETH DAY OF MAY 2013

TO THE DEPARTMENT OF PHYSICS AND ENGINEERING PHYSICS

IN PARTIAL FULFILLMENT OF THE REQUIRMENTS

OF THE SCHOOL OF SCIENCE AND ENGINEERING

OF TULANE UNIVERSITY

FOR THE DEGREE

OF

DOCTOR OF PHILOSOPHY

BY

__________________________

YUAN FANG

APPROVED: ______________________________

JOHN P. PERDEW, Ph.D. DIRECTOR

_______________________________

ADRIENN RUZSINSZKY, Ph.D.

_______________________________

JIANG WEI, Ph.D.

_______________________________

LEV KAPLAN, Ph.D.

Abstract

Weak bonds such as hydrogen bond, pi-pi stacking and van der Waals

interaction are much weaker in the strength but play a more important role for the

existence of various lives. For example, they are the major intermolecular interactions

in the liquid and solid structure of water and determine the 3 dimensional structure of

protein and DNA, which are the crucial organic molecules in lives. As a result,

studying these weak bonds can lead to the better understanding of fundamental

knowledge of lives.

Kohn-Sham (K-S) Density Functional Theory (DFT) is an accurate and effect

way to investigate the fundamental properties for many-body systems, in which, only

the exchange-correlation energy as a functional of electron density need to be

approximated. However, weak interaction system is still a challenge problem for

KS-DFT. In this dissertation work, several standard density functionals are used to

study these weak interactions in the solid state structure ice as long as nucleic bases

molecules in the biologic system. It is found that the hydrogen bond can be well

described by most semilocal functionals: the mismatch problem of ice Ih and AgI for

GGA functional can be solved by using the higher level meta-GGA functionals and

the binding length and energy between nucleic bases in DNA can be well described.

However, the more accurate dispersion correction is strongly needed for van der

Waals interactions and pi stacking for super-high pressure ice phases and large size

biologic molecules, where van der Waals interaction takes major role. Finally, the

basic structural properties of various phases of ice and DNA can be understood based

on the investigation with appropriate functionals.

HYDROGEN BOND, PI-PI STACKING, AND VAN DER WAALS INTERACTIONS

INVESTIGATED WITH DENSITY FUNCTIONAL THEORY

A DISSERTATION

SUBMITED ON THE THIRTIETH DAY OF MAY 2013

TO THE DEPARTMENT OF PHYSICS AND ENGINEERING PHYSICS

IN PARTIAL FULFILLMENT OF THE REQUIRMENTS

OF THE SCHOOL OF SCIENCE AND ENGINEERING

OF TULANE UNIVERSITY

FOR THE DEGREE

OF

DOCTOR OF PHILOSOPHY

BY

__________________________

YUAN FANG

APPROVED: ______________________________

JOHN P. PERDEW, Ph.D. DIRECTOR

_______________________________

ADRIENN RUZSINSZKY, Ph.D.

_______________________________

JIANG WEI, Ph.D.

_______________________________

LEV KAPLAN, Ph.D.

ii

Acknowledgments

I would never have been able to finish my dissertation without the guidance

of my committee members, help from my group members and friends, and support

from the physics department, my family and wife.

First of all, I would like to express my deepest gratitude to my advisor, Dr.

John Perdew, for his excellent guidance, support, and remarkable patience. I would

like to thank my committee members, Lev Kaplan, Ph. D., Jiang Wei, Ph. D., Adrienn

Ruzsinszky, Ph. D., for their helpful insights for my dissertation. I would also like to

thank all my group members, Dr. Perdew, Dr. Ruzsinszky, Dr. Jianmin Tao, Dr.

Jianwei Sun, Bing Xiao, Dr. Pan Hao, for providing me with an excellent atmosphere

for doing research and all the helpful suggestions and discussions. The work in this

dissertation was undertaken mostly during a teaching fellowship at the physics and

engineering physics department. I gratefully acknowledge the support.

I would like to thank all my friends in New Orleans for the entire

accompaniment during the last six years.

An especially thank for my parents, Xin Fang and Shuyu Zhu, for all the

encouragement and education.

Finally, I would like to thank my wife, Dr. Shanshan Shen, for her constant

friendship, love, and support for all these ten years.

iii

Table of Contents

Acknowledgements ii

Table of Contents iii

I. Introduction: The bonds of life 1

II. Theoretical information:

Wavefunction and density functional theory,

approximations, and computer codes 9

2.1 Many-body Schrödinger equation problem 9

2.2 Wavefunction based approximation 11

2.3 Density functional theory 12

2.3.1 History of density functional theory 12

2.3.2 Approximation for exchange-correlation functionals 16

2.3.3 Climbing the "Jacob's ladder" of density functional 17

2.3.4 van der Waals Correction Functionals 29

2.4 Theory to Practical Calculation 31

2.4.1 The Vienna Ab initio simulation package (VASP) and PAW

method 32

2.4.2 Gaussian and Gaussian Type Orbitals 37

iv

III. Ice phases under ambient and high pressure: Insights for

density functional theory 40

3.1 Introduction 40

3.1.1 Interactions inside ice 40

3.1.2 Density functional theory for solid ice 42

3.2 Computational details 45

3.3 Results and Discussion 47

3.3.1 Lattice mismatch challenge for ice-Ih and β-AgI 47

3.3.2 Sublimation energy for ice phases under ambient and high

pressure 52

3.3.3 Transition pressure between ice phases 56

3.4 Liquid water 61

3.5 Conclusions 61

IV. What forces twist the Deoxyribonucleic Acid?

Interactions between nucleic bases with density

functional theory 63

4.1 Introduction 63

4.1.1 Deoxyribonucleic acid structure and nucleic bases 63

4.1.2 twist of DNA and DNA base pair steps 64

4.1.3 Ab initio methods for nucleic bases 67

v

4.2 Computational details 69

4.3 Results and Discussion 72

4.3.1 Hydrogen bond interaction between Watson-Crick pairs 73

4.3.1.1 Bond length and bonding energy 74

4.3.1.2 The contribution of vdW dispersion force for bonding

Between base pairs 76

4.3.2 pi-pi stacking interactions between stacked bases 78

4.3.2.1 Benzene dimer stacking 79

4.3.2.2 Rise and potential energy surface (PES) for stacked nucleic

bases 83

4.3.3 Stacking interactions for stacked nucleic pairs and possible

reason for the twist of DNA from the view of PES 91

4.4 Conclusion 94

References 96

1

1

Chapter I

Introduction: The bonds of life

The original spark of life may have begun in a warm little pond, with all sorts of

ammonia and phosphoric salts, light, heat, electricity, etc. present, so that a protein

compound was chemically formed ready to undergo still more complex changes.

- Charles Darwin[1], 1871

Although it is still under debate how the first self-replicating molecule

formed, it is well accepted that these RNA, DNA, and protein molecules evolved to

the first living cells in a warm pond, the ancient ocean. While DNA is the organic

molecule encoding the genetic instructions used in the development and functioning

of almost all known life, water, an inorganic molecule, is considered to be the most

important fundamental requisite for life. In fact, up to 70% of the human

body consists of water. However, it is not until the last one hundred years that

scientists have discovered some fundamental properties of water and DNA molecules,

and many questions remain to be answered.

Water covers 71% of the earth’s surface and often co-exists in its solid,

liquid and gaseous states. Only in the 19th

century was the composition of the water

molecule proposed to be H2O[2], with one oxygen atom and two hydrogen atoms

connected by covalent bonds in a polar molecule. Water exhibits some special

physical and chemical properties: Water is in the liquid state at room temperature and

transits to the solid phase when the temperature is below 273K[3], which is common

2

2

during winter on most parts of the earth’s surface; water’s density changes with

temperature with the maximum occurring around 277K, and the density decreases by

~10% when frozen to solid ice; water is a good polar solvent, often referred to as a

universal solvent in chemistry; some organic molecules like protein and DNA are

easily dissolved in water, etc. To find out the mystery of water, for decades

experimental chemists and physicists tried to figure out the basic structure of liquid

and solid water. With the development of X-ray and neutron diffraction technology,

the structure of natural ice was confirmed in the middle of 20th

century[3,4], while

that of liquid water is still under debate. The crucial factor to study water is the

understanding of the intermolecular interactions, principally hydrogen bonds. A

hydrogen bond in water is the electromagnetic attractive interaction of a

slightly-positive hydrogen atom in one molecule to a slightly-negative oxygen atom in

an adjacent water molecule. The strength of the hydrogen bond is about 2-10 kcal/mol

which is similar to the thermal energy. This makes water stay liquid under room

temperature and easy to freeze. The exact number of hydrogen bonds in liquid water

fluctuates with time and depends on the temperature, which leads to the change of

density. Unlike in solid ice, hydrogen bonds in water only have 1-20 ps lifetime with

bond breaking and rebuilding in 0.1ps, which leads to the difficulty of measuring the

structure of liquid water. Hydrogen bonds also play an important role when water

interacts with other structures, making it a good solvent. In particular, the existences

of hydrogen bonds in organic molecules like protein and DNA, which are also

3

3

considered in this work, make them soluble in water.

Deoxyribonucleic acid (DNA)[5] is a molecule encoding

the genetic instructions and famous for its double helix structure first discovered by

James D. Watson and Francis Crick in 1953. The common B-DNA double helix has

double-stranded helices, consisting of two long backbones called grooves made of

alternating sugars and phosphate groups, with the nucleic bases attached to the sugars.

A nucleotide is building block of DNA which consists of one pair of linked bases plus

the attached part of the backbone. Though there are billions of nucleotides in one

double DNA chain, only four kinds of nucleic bases exist in DNA: adenine (A),

cytosine (C), guanine (G) and thymine (T), and their order stores the genetic

information following the base pair rules: A pairs with T, G pairs with C. The double

helix makes one complete turn about its axis every 10.4-10.5 base pairs in solution,

which means that there is a twist of about 36 degrees between neighboring

hydrogen-bonded pairs of bases. This twist is formed and stabilized by various forces.

Two of them are named the hydrogen bond which is between the nucleic bases on

different backbones and the pi-pi interaction between nucleic bases on the same

backbone. The interaction is a weak interaction between aromatic rings with energy of

2-10 kcal/mol, which contributes at least as much as the hydrogen bond for the

stability between bases and also influences the strength of hydrogen bond. However,

while the nature of the hydrogen bond is well researched, this interaction is still

actively debated in the literature since the 1990s, because it is a combination of van

4

4

der Waals dispersion forces, electrostatic interactions and Pauli repulsions.

Another important weak interaction is the van der Waals interaction, the sum

of the attractive forces between molecules resulting from fluctuating

instantaneously-induced dipoles. Typical energies for vdW interactions are around

0.5-1 kcal/mol, much smaller than covalent bonds with energies of 50-200 kcal/mol.

However, since this value is comparable with the strength of the hydrogen bond and

pi-pi interaction, the influence of vdW interactions must be taken into consideration

when dealing with weakly-bound solids or molecules. Recent findings show that vdW

interactions play an important role in the structure of liquid water and high-pressure

ice and in the 3D stability of a protein molecule. In experiment, the measurements for

such weak interactions are difficult. In fact, not until 2012 was made the first direct

measurement of the strength of the van der Waals force for a single organic molecule

bound to a metal surface[6].

High-accuracy theoretical simulation methods for all these weak interactions

are needed. For hundreds of years, experiment was the only way to explore the

mystery of life and the molecules of life. Although many discoveries were made,

scientists suffered from the technical limits of theoretical methods and experimental

equipment, especially on atomic scales. Thanks to the invention of computers in 1946,

scientific simulations with computers became the second way to understand the world.

Nowadays, with high-performance super computers and efficient algorithms,

scientific simulations have been used not only to verify the experimental findings but

5

5

also to direct the experiments: Increasingly, material data bases constructed by

computation can be used to predict new materials with desired properties.

Although computational techniques were dramatically revolutionized during

the last half century, atomic-scale computation for large systems is still a huge

challenge for computer modeling. The requirements for various simulation methods

are both accuracy and efficiency. All these methods can be grouped into wave

function and density functional approximations, which will be explained in detail later

in this dissertation. Even though, in some situations, wave function methods work

more accurately than density functional theory (DFT)[7], their computational cost is

significantly larger than DFT, and so these methods usually can only be used for

system with several tens electrons. At the other hand, DFT is capable of dealing with

systems with more than hundreds of electrons. With the development of new DFT

approximations, the accuracy problem is being solved. Thus, DFT is now the most

commonly-used ab initio method in both condensed matter physics and quantum

chemistry.

Nowadays, there are hundreds of different density functionals. By how

much information about the density they take into their xc energy approximations, we

can sort them as local, semilocal, and nonlocal functionals. With the development of

GGA functionals in the 1990s, DFT was considered accurate enough for the

calculation of atoms, molecules and solids. The famous GGA PBE[8] is cited more

than 30,000 times and is widely used in condensed matter physics, while the hybrid

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6

functional B3LYP[9,10] is widely used in computational chemistry.

FIG. 1-1: Number of papers with topic including PBE, B3LYP, and other DFT

functionals[11].

However, there are still difficulties in using GGA-DFT[12] to properly

describe intermolecular interactions like van der Waals forces, as well as transition

states for chemical reactions, global potential energy surfaces, some other strongly

correlated systems, and the band gaps in semiconductors. To solve these problems,

many improved functional approximations have been invented. As the highest level of

the semilocal functionals, the meta-GGA is a natural way to improve accuracy further

by making use of additional semilocal information. Meta-GGAs have yielded better

results than GGAs for covalent, ionic, and metallic solids, including atomization

energy, lattice constant, bulk modulus, and cohesive energy[13]. Although the GGA

7

7

functionals were already widely tested for hydrogen bonds and organic molecules, the

performance of meta-GGA functionals for such system is still not well studied,

especially for some newly developed functionals. This work aims to establish the

ability of the meta-GGA to describe weakly bonded structures.

In chapter three of this dissertation, the interactions in the water solid

structures are studied with several density functionals on different levels. At first, the

easier case of ambient-condition solid phase ice-Ih is studied for a lattice mismatch

problem, to learn how these xc functionals perform when hydrogen bonds play the

major role. After that, in view of the importance of vdW forces at high pressure[14],

the properties of two high-pressure phases of ice are studied. By comparing the results

of all these DFT functionals to the experimental data, one can explore their

advantages and shortcomings for systems with a combination of hydrogen bonds and

vdW interactions.

In chapter four, the work for the DNA nucleic base pairs is shown. It is well

known that the interactions of nucleic bases coming from the hydrogen bond and pi-pi

stacking interactions. To isolate the influence of them, first the bonding energy and

bond length which are between nucleic bases on the same rung were studied, then the

rise (vertical separation between stacked bases) and potential energy surface was

computed as a function of the twist angle between two stacked bases. By comparing

to those found by quantum chemical wave function based methods, the performance

of different functionals can be understood. Finally, the preferred configurations for

8

8

possible DNA Watson-Crick bases pairs (with four nucleic bases) are studied at a

preliminary level. This problem would be very hard to study with quantum chemical

methods.

All the interactions studied here are much weaker than those of common

chemical bonds, but should not be neglected. Considering their importance in

chemical reactions, and to life, a better description of this weak interaction is needed.

The major question to be addressed here is how all these density functionals’ perform

for systems with hydrogen bonds is or with a combination of several types of weak

bonds including van der Waals as well as hydrogen bonds, and which functional

should be favored in future studies. Before answering these questions, some

theoretical concepts must be introduced. In the next chapter, basic density functional

theory and density functional approximations and computer codes are briefly

discussed, with special focus on the meta-GGA functional.

9

9

Chapter II

Theoretical Information: Wavefunction and

Density Functional Theory, approximations, and

computer codes

2.1 Many-body Schrödinger equation problem

Condensed matter physics is a branch of physics that deals with the physical

properties of condensed phases of matter. It overlaps with chemistry, materials science,

and nanotechnology, and relates closely to atomic physics and biophysics. Theoretical

condensed matter physics attempts to understand and manipulate the properties of

materials by studying the interactions of nuclei and electrons from fundamental

physical principles such as quantum and statistical mechanics. The key way to get

information of the ground-state properties is to solve the many-body time-independent

Schrödinger equation[15] for a system containing N electrons and M nuclei shown as

Eq. [2-1],

[2-1]

with the Hamiltonian

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10

[2-2]

where is the ratio of the mass of nucleus A to that of an electron; is the

atomic number of nucleus A. R and r are the coordinates of nuclei and electrons,

respectively.

When N=M=1, this is a typical hydrogen-atom Schrödinger equation. When

new nuclei and electrons are added, this equation becomes too complex to be

mathematically solved because all the correlation between the particles needs to be

considered. Generally, for a condensed matter physics problem, we may have at least

several nuclei and tens of electrons in a repeating unit cell, a problem which cannot be

perfectly solved. Thus, some reasonable approximations must be made to simplify

this many-body Schrödinger equation problem.

The first approximation is the Born–Oppenheimer (BO) approximation[16]

proposed in 1927. In the BO approximation, the nuclei are treated as fixed due to the

high ratio between nuclear and electronic masses. Then the nuclei kinetic term and

nuclei repulsion term can be removed and the many-body Hamiltonian reduces to a

three-term many-electron Hamiltonian with form in Eq. [2-3].

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11

[2-3]

Then the problem is to solve the electronic Schrödinger equation to get the electronic

wave function and energy. After that, the total energy can be found by adding the

nuclear repulsion energy to the electronic energy.

[2-4]

However, because of the interaction between electrons, this equation is still hard to

solve, and further approximation is needed. All these approximations can be sorted as

wavefunction based approximations and density functional theory.

2.2 Wavefunction based approximation

Wavefunction based approximations are not used in this work, but since some

MP2 results are used to benchmark DFT calculations later, some basic information

about them will be briefly introduced first.

A major aim of wavefunction based methods is to find the many electron

wavefunction . First is the Hartree-Fock (HF) approximation, which assumes that

the exact wavefunction of the system can be approximated by a single Slater

determinant. The HF approximation is the central starting point for many other

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12

methods, and is even involved in the hybrid density functionals. However, neglecting

electron correlation in HF may lead to large deviations from experimental results,

especially for energy. So a set of methods, referred to as post Hartree-Fock, have been

developed to improve the HF approximation by adding electron correlation. These

methods, including the Coupled Cluster method (CC), Møller–Plesset perturbation

theory (MP2), Quadratic configuration interaction (QCI), Quantum chemistry

composite methods (QC), and so on, can give more accurate results than HF and are

often used as reliable computational benchmarks[17] for other methods like DFT,

particularly in the situation where experiment measurements are hard to perform. The

accuracy with these methods comes with significant increasing computational cost, so

they are not fit for large complex systems and molecule dynamics simulations. A

useful compromise between computational accuracy and efficiency can be achieved

by density functional theory (DFT).

2.3 Density functional theory

Unlike the wavefunction based methods, the density functional theory invokes a

functional of the electron density to find the ground-state total energy and density.

Mathematically, a functional is a rule that maps a function like the electron density

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13

[2-5]

into a number like the total energy. By doing this, DFT reduces the 3N-dimensional

many-electron problem to a 3-dimensional problem, which can be practically solved

for much larger systems. Nowadays, no other method can achieve comparable

accuracy to DFT at the same computing cost.

2.3.1 History of density functional theory

Back in the 1920s, Thomas and Fermi first approximated the energy in terms

of the electron density. The Thomas-Fermi Model[18,19] was simple in form, and not

very accurate, but it provided the conceptual root for density functional theory: the

electron density can determine the energy and other properties of the ground state.

In the 1960s, the firm theoretical base for DFT was constructed by the two

Hohenberg-Kohn theorems[20]. In the B-O approximation, the external potential

determines the kinetic energy of electrons, the interaction energy between electrons,

and the electron density. Hohenberg and Kohn proved that the inverse relationship

also exists: the electron density can uniquely determine the external potential ,

and thus other quantities. The first Hohenberg-Kohn theorem states that the ground

state properties of a many-electron system are determined in principle by the electron

density. Then the second H–K theorem defines the energy functional for the system

and proves that the correct ground state electron density can be found by using the

variational principle. The total energy can be expressed as a functional of density, and

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14

the exact ground state density is the one that minimizes the total energy.

[2-6]

If we know the form of the functional , then in a given external potential we

can find the ground state density and energy by minimizing the total energy.

However, the exact functional is not known in any computable form, so

approximations are still needed.

In 1965, the discovery of the Kohn-Sham equations[21] made the density

functional theory into a practical and realistic computing method. Within this

framework of Kohn–Sham density functional theory (KS-DFT), the

intractable many-body problem of interacting electrons in a static external potential is

reduced to a tractable problem of non-interacting electrons moving in an

effective potential, typically denoted as , called the Kohn-Sham potential. In the

Kohn-Sham theorem, the functional is expressed as

[2-7]

where is the Kohn-Sham kinetic energy which is the kinetic energy of all

the non-interacting electrons, (hartree energy) is the Coulomb energy

15

15

between the electrons, and is the exchange-correlation energy. Then the

Kohn-Sham potential can be found from the functional derivative

.

[2-8]

In the Kohn-Sham system, the electrons are non-interacting; the one-electron

Kohn-Sham Schrödinger equation can be written as

[2-9]

where is the orbital energy of the corresponding Kohn–Sham one-electron orbital

and the density of the N-electron system is defined as

Since the Kohn-Sham potential [2-8] depends on the electron density which is

defined above as Eq. [2-5], this Schrödinger equation must be solved self-consistently.

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16

FIG.2-1 Self-consistent process

One typical self-consistent procedure is shown as FIG. 2-1. Generally, starting

with an initial electron density, is built. Then by solving the Kohn-Sham

equations, a set of Kohn-Sham orbital can be found, which give the new electron

density. These calculations are repeated until a specified level of convergence

requirement is achieved. After that, with the true electron density, the total energy can

be computed. The Kohn-Sham method is quite efficient since non-interacting systems

are relatively easy to solve as the wave-function can be represented as a Slater

determinant of orbitals and the kinetic energy functional of such a system is known

exactly. However, the term in Kohn-Sham energy is not known exactly. To

find a reasonably accurate approximation for is the major task of Kohn-Sham

density functional theory.

17

17

2.3.2 Approximation for exchange-correlation functionals

The first attempt or Hartree approximation simply ignored the exchange-correlation

energy term in the Kohn-Sham theory and turned out to imply drastically too-weak

bonds between atoms and drastically too-long bond lengths and lattice constants. This

indicates that, even when it is a relatively small part of the total energy, the

exchange-correlation energy is a crucial part of the interactions between atoms, and it

is therefore sometimes referred to as "nature’s glue"[22]. As a result, for accurate use

of KS-DFT, must be included, however, since contains all

the electron-electron interactions in the real system which are excluded in the Hartree

approximation, the exact form or even an accurate-enough approximation are hard to

build. It took ten years for the completion of the first practical approximation, the

local spin density approximation (LSDA), and another ten or twenty years for the

completion of the generalized gradient approximation (GGA), which is almost

accurate enough for computation in both solid state physics and quantum chemistry.

Basically, there are two ways to construct the approximate functionals: non-empirical

and empirical. Non-empirical density functionals are constructed to satisfy some exact

constraints on ; they are not fitted to certain real results so they can be

universal for all systems. At the other hand, (semi) empirical functionals include

parameters which are fitted to other computed or experimental results. These

functionals may be more accurate than non-empirical functionals inside their fitting

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18

range but can fail outside it. Today, there is a long list of different kinds of density

functional. An efficient way to sort them is using the "Jacob's ladder[23]" which

was proposed by Perdew and Schmidt in 2002.

2.3.3 Climbing the "Jacob's ladder" of density functional

Generally speaking, most exchange-correlation approximation can be expressed as

[2-10]

where n is the electron density, is the gradient of the density and is the kinetic

energy density. Based on the complexity of density information that is taken into

account, these approximations can be sorted into five levels as shown in FIG. Like the

biblical Jacob's ladder that the Patriarch Jacob dreamed about, it is a ladder that links

the earth to heaven. The DFT Jacob's ladder bridges the Hartree world and the

chemical-accuracy heaven. Climbing the ladder from one rung to the one above, a

new argument or ingredient of the density is added to achieve higher accuracy, at

some increase of computing cost. Noting that the increase of accuracy is a statistical

result, the performances of various functionals depend on the problem they treat,

especially for the empirical functionals that are fitted for certain systems and

properties.

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19

FIG.2-2: Jacob's ladder

Next we will discuss some basic functionals on each rung, especially those

used in the later chapters.

(i) Local Spin Density Approximation

As the earliest and simplest attempt designed in the 1960s and 1970s, LSDA

depends only upon the value of the electronic density at each point in space and

excludes other terms like the derivatives of the density. Under this approximation, the

exchange-correlation energy of the system can be expressed as

20

20

[2-11]

where are the spin-up and spin-down density, is the total electron

density, and

is the exchange-correlation energy per electron for an

interacting uniform electron gas with spin density .

Generally, the exchange-correlation is the sum of the exchange energy

and the correlation energy . The exchange energy functional for LSDA is

analytically known. In the spin-unpolarized case,

[2-12]

The analytic form for the LSDA correlation energy functional is not known except in

low and high density limits. So the LDSA is a controlled interpolation between

these two limits, and this accounts for the difference between various LSDA

functionals such as VWN, PZ81, etc[24--26].

As suggested by its method of construction, LSDA is only exact or accurate

for systems with uniform density or slowly-varying density such as in some solids. In

fact, during the 1970s and 1980s, LSDA-DFT was more successfully used in

condensed matter physics than quantum chemistry, and worked surprisingly well for

many solids. This can partly be explained by the fact that that the

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21

exchange-correlation hole of LSDA satisfies exact hole constraints, and partly by an

understood error cancellation between LSDA exchange and correlation energies. In

LSDA, the exchange energy is typically underestimated by 10% and correlation

energy is overestimated by 100%. Then the total energy is typically too high. LSDA

shows 1%-3% underestimation of lattice constant, and its atomization energy for a

molecule is typically too large[27]. Besides, the description for weak bonds is bad for

LSDA: It overestimates the short-range part[28,29] and underestimates the long-range

vdW part. However, the cancellation of error in LSDA sometimes can lead to

surprising results such as better surface energy for a simple metal than GGAs. As the

first rung of the KS DFT, LSDA gives the starting point for the construction of

higher-level functionals.

(ii) Generalized Gradient Approximation (GGA)

To improve the performance of LSDA and work beyond the uniform-density

limit, a logical idea is to construct a new functional to include the derivatives of

density. For example, the gradient expansion approximation (GEA) tries to add the

energy corrections with a gradient expansion to second- or fourth-order in the

gradients. But these attempts show little improvement over LSDA, and are even

worse[30] for the reason that truncation of the gradient expansion violates

exchange-correlation hole constraints.

To get around this problem, the exchange-correlation energy can be expressed

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22

in a more generalized form using the density and its gradient, which is known as the

generalized gradient approximation (GGA)[12].

[2-13]

The construction of can proceed in various ways. One route to the PW91 and

PBE GGAs used here is to start with the second-order gradient expansion for the

exchange-correlation hole, then introduce sharp real-space cutoffs to satisfy the exact

hole constraints. Generally, the GGA exchange-correlation energy can be defined as

[2-14]

where is the enhancement factor for the uniform exchange, is the local Seitz

radius (n=3/4π ), is the relative spin polarization, and

is a dimensionless density gradient. The choice of the mathematical form

for makes the difference between GGAs. For example, in the most widely-used

GGA functional, the Perdew-Burke-Ernzerhof (PBE)[8], a simple is defined as

[2-15]

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23

where are two non-empirical coefficients. And the can have a more

complex form.

Compared to LSDA, GGA PBE can predict much more accurate atomization energies

for molecules, so DFT expanded its application to quantum chemistry after the birth

of GGA. However, a problem with GGAs is that none of them can achieve an

accurate performance for both atomic/molecular energy and for solid/surface energy.

GGAs which aim to describe atoms and molecules will fail in the solid and vice verse.

PBE usually overestimates the lattice constants of solids by 1%-2% and

underestimates the surface energy of a metal, while PBEsol[31], which is restores the

density-gradient expansion for exchange in solids and surfaces, improves equilibrium

properties for solids and surfaces, at the cost of accuracy for atoms and molecules.

This problem needs to be solved with new approximations which can be applied with

accurate results in both molecules and solids. That is one of the motivations for the

construction for the meta-GGA functionals.

(iii) meta-GGAs

Meta- is a prefix to indicate a concept used to complete or add to an earlier

concept. In this sense, meta-GGAs, the third rung on the ladder, are constructed to

improve the performance of GGAs by taking into consideration additional

information about the electron density like the Laplacian of the density or the

kinetic energy density. The typical formula for meta-GGA xc energy can be expressed

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as

[2-16]

where are the electron density and gradient of the density included in

GGAs, and , the new information included in the meta-GGAs, is the kinetic energy

density for the occupied KS orbitals .

[2-17]

Although the idea of meta-GGA arose in the 1990s, it is not until recent years that

practical meta-GGAs have been constructed. Subsequent research showed that they

improve over GGAs in many situations, including the hydrogen bond in water clusters,

and this is the reason we seek to understand their performance for solids with

hydrogen bonds and other weak interactions. As for GGAs, meta-GGAs can be

constructed non-empirically or empirically. John. Perdew leads the non-empirical

effort by constructing TPSS[32], revTPSS[13], etc, while the Minnesota

Functionals[33] constructed by Don Truhlar are widely-used semi-empirical

meta-GGAs.

Non-empirical functionals TPSS and revTPSS:

The starting point for TPSS is the original version Perdew-Kurth-Zupan-Blaha

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(PKZB)[34] meta-GGA, which poorly describes the bond length and hydrogen bond.

The key idea in TPSS[32] is to restore PBE-like GGA behavior at large reduced

density gradients, where the constraints added in PKZB do not apply. Somewhat as in

GGAs, the exchange energy of meta-GGA can be expressed as

[2-18]

in which again

is the exchange energy density for uniform electron gas, and

is the meta-GGA exchange enhancement factor with two

dimensionless parameters p and z. and , where

is the

Weizsaecker kinetic energy density, and is the orbital kinetic density defined

earilier. And the analytic form of is constructed to satisfy the one- or

two-electron density constraints. The improvement in the exchange part makes TPSS

better than earlier PKZB by shortening the bond length and improving the energy for

hydrogen bonds. Besides, TPSS also implies good results for surface energy and

atomization energy, but lattice constants only slightly better than those of the PBE

GGA. revTPSS was constructed to further improve the lattice constants. The

construction principle for revTPSS is similar to that of PBEsol, restoring the

second-order gradient expansion for exchange over a wide range of densities to

achieve better results for lattice constant, at the same time keeping the good

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26

performance of TPSS for surface and atomization energy. Recently testing show that

this new revised version meta-GGA works well for many systems and may "become a

workhorse semi-local density functional for both condensed matter physics and

quantum chemistry."[13]

Empirical functional M06L:

Except for the TPSS-like functionals, which are constructed by satisfying

several physical constraints, some other functionals are constructed by fitting the

parameters to real-system data, such as the widely used Minnesota functionals family,

which includes tens of meta-GGA and hybrid functionals. The meta-GGA M06L[33]

is one of the most successful and widely accepted functionals. The exchange energy

in M06L can also be expressed in the similar form

[2-19]

where ,

is the exchange energy density for LSDA and PBE,

, are two functions of the kinetic energy density. Unlike TPSS, these

two functionals are built by fitting the parameters to a molecular database. As a result,

M06L performs quite well for molecules, even better than several hybrid functionals.

However, the performance of M06L outside its fitting range, as in solids, can be poor,

and the huge number of fitting parameters in M06L may cause some calculation

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problem for extended systems.

FIG. 2-3: List of optimized parameters used in the M06L meta-GGA[33]

MGGA_MS:

MGGA_MS (meta-GGA made simple)[35] is a newly developed semi-local

functional. The dimensionless inhomogeneity parameter α, which is defined as

is introduced into the meta-GGA exchange enhancement factor

to characterize the extent of orbital overlap. By adding this exchange functional with a

variant of the PBE correlation functional, the MGGA_MS xc energy is built. This new

functional performs better than GGA PBE for atoms, molecule, solids and comparable

with the meta-GGA revTPSS, but with a much simpler form. Besides, since α can

recognize different degrees of orbital overlap, it can distinguish the weak bonds from

the strong binds. In this case, MGGA_MS has the ability to capture at least part of

weak interactions, which is a challenge for the conventional GGA and meta-GGA

functionals.

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FIG. 2-4: The binding curve of Ar2, which is a typical molecule bound by a

noncovalent bond[36]

(iv) Hybrid Functionals

Hybrid functionals mix the exact exchange energy from the HF method

with the exchange-correlation energy from another method such as DFT, which is

why they are called "hybrids". A typical formula for a hybrid-GGA

exchange-correlation energy is

[2-20]

where is the exact exchange energy,

and are the GGA exchange

and correlation energies, and "a" is the mixing coefficient. Different hybrid

functionals may employ different GGA xc energy and mixing coefficients.

For example, B3LYP[9,10] is a hybrid popularly used in the chemistry. It

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uses the exchange energy of the B88[37] GGA and the correlation energy of LYP[38].

[2-21]

while the PBE0[39] using both exchange and correlation energy of GGA PBE is more

widely-used in condensed matter physics due to its lack of empirical fitting

parameters.

[2-22]

Recently, meta-GGA exchange-correlation functionals have also been used to construct

hybrid meta-GGA functionals such as Minnesota 06 family of functionals.

Hybrid functionals are fully nonlocal and tend to yield better performance than

semi-local functionals, although they also increase computing time due to the large

computational requirement for the exact HF exchange, so they are used more to treat

the problem of molecules. Furthermore, the performance of various hybrid functionals

strongly depends on the applying system; B3LYP works well for molecule, but fails in

metals and other extended systems.

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30

(v) Random Phase Approximation

The RPA was introduced back in the 1950s to estimate the correlation energy of

the uniform electron gas[40]. The “RPA in a density functional context” was

introduced later by Langreth and Perdew[41]. It took more decades before the RPA

could be practically used in the KS-DFT. Generally, the RPA xc energy can be

expressed as the summation of exact exchange energy and RPA correlation energy:

[2-23]

In RPA, the long-range dispersion interactions are well described, but the short-range

correlation is overestimated. So a short-range semilocal correction can be added to

RPA, yielding RPA+[42,43]. A major problem with RPA is the computational cost,

maybe 100 times greater than that of GGA.

2.3.4 van der Waals Correction Functionals

The classical semilocal functionals such as LSDA, PBE, PBEsol, TPSS, and

revTPSS are not fully nonlocal, and even the hybrid functionals with the nonlocal HF

exchange mostly cannot yield good results for systems where the long-range van der

Waals interactions need to be taken into consideration, as in many biological molecules.

Using RPA can solve this, but with a significant increase of computing cost. An

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31

alternative is to add a correction for the dispersion energy to the DFT xc energies,

which can be expressed as a power series[44]:

[2-24]

where R is the distance between two particles, is the coefficient which describes the

dipole-dipole interaction, and is the coefficient which describes the

dipole-quadrupole interaction. In large-separation situations, the first term usually

plays the major role and is easy to include. Although adding the following ,

terms may decrease the error, they are hard to get. So the practical step is to find a

reasonable approximation for the coefficient . Nowadays, there are many

methods[45--49] designed to provide a long-range vdW correction to semilocal

functionals, and we will use several of them in later chapters.

One is the DFT-D2[47] approach of Grimme, in which the van der Waals

interactions were described via a simple pair-wise force field and added to the

conventional Kohn-Sham DFT energy. The dispersion potential energy for periodic

systems is defined as

[2-25]

where the dispersion energy is defined as

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32

where is the number of all atoms, is a global scaling factor,

denotes the dispersion coefficient for the atom pair i and j and the damping functions

is defined as

Because of the simplicity and low computational cost, this pairwise correction is

widely used and implies quite good results. However, there are still some

shortcomings because of the neglect of non-pairwise effects. The coefficient is

mostly taken from experiment.

Recently, a new version called DFT+D3[50], adding the term to the

dispersion energy, has been constructed but is not included in this work.

Second is the vdW-DF[48] method proposed by Dion et al. It is a non-local

correlation functional added to a semilocal xc energy to account for dispersion

interactions:

[2-26]

where is the exchange energy of a certain GGA functional,

is the local

spin-density approximation (LDA) to the correlation energy and is the

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33

approximate nonlocal energy term. optB88-vdW and optB86b-vdW used in our work

are basing on the vdW-DF approach with the Becke88 and Becke86 exchange

functional optimized for the correlation part. Early work shows that the performance

of this kind of functional may depend on the GGA functional included in it.

2.4 Theory to Practical Calculation

As mentioned before, the KS equations must be solved self-consistently to

get the ground state electron density; the process can only be performed by the

computer or super computer. In 2006, more than five million hours of computer time

was used by DFT simulation at NERSC in the Lawrence Livermore National

Laboratory. Therefore, accurate and efficient codes for the DFT calculations are

highly needed. There are now hundreds of codes written for DFT. The major

difference among them is the choice of the type of basis sets for the expansion of the

KS orbitals, including plane waves, localized atomic orbitals (LCAO), and augmented

plane waves. Each of them has advantages and disadvantages and the best choice

depends on the system treated. Generally speaking, for finite systems such as atoms,

molecules and clusters, LCAO usually works well while the plane wave method

works well for extended periodic systems like solids. In the following, two codes used

in our work are briefly introduced.

2.4.1 The Vienna Ab initio simulation package (VASP) and PAW

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34

method

The Vienna Ab initio Simulation Package (VASP)[51,52] is a computer

program for atomic- scale materials. It can compute an approximate solution for the

many body Schrödinger equation within the KS-DFT and HF approximation using a

plane-wave basis set and the projector-augmented-wave (PAW) method. Thus, it is

more suitable for periodic solid structures. Today, VASP is used by more than 1400

research groups in academia and industry worldwide.

Unlike in isolated atoms or molecules, in an extended system such as a solid

structure, which usually consists of a huge number of electrons, it is impossible to

calculate wavefunctions., Instead, the solution is treat the solid as a periodic structure,

which is built by repeating a small part of the solid, called the unit cell, infinitely in

all three dimensions. For a so- called periodic solid structure, the plane-wave basis set

is widely used because of Bloch's theorem which implies that in a periodic system the

KS or HF orbital can be written as the product of a plan wave term and a

periodic factor ,where

[2-27]

Since

, where R is a Bravais lattice vector, it suffices to calculate

in only one unit cell. Next the periodic term can be expanded as a series of

plane waves with wave vectors G, the reciprocal lattice vector of the periodic crystal.

[2-28]

Then the orbital can be expressed as

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35

[2-29]

Finally, the Kohn-Sham equation can be rewritten in the plane-wave basis set form

[2-30]

where

is the plane-wave kinetic energy, and is the electronic energy.

Plane-wave cutoff and k-points:

In the above equation, an infinite number of G’s need to be summed in principle

which is impossible for real calculation, while in practice the actual number of G's is

determined by the kinetic energy cutoff . This approximation is reasonable

because the plane waves with smaller kinetic energy are much more important than

the others, so only the plane waves with the kinetic energy satisfying

are taken into the calculation. Using the energy cutoff may decrease the

accuracy for the total energy. However, a convergence test which finds the maximum

value of to make the total energy converge can minimize the error.

The total energy is evaluated from the contributions of each occupied

electronic state at every wave vector (or k-point) in the solid. However, in a periodic

structure, the Irreducible Brillion zone (IBZ) in the reciprocal space, which is a

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36

fraction of the complete unit cell of the BZ, is sufficient to determine all the

information for the crystal. Increasing the k-points can improve the accuracy, tending

to the true total energy while the computing cost increases linearly. Consequently, a

convergence test for k-points is also needed before making the calculation for each

system.

Pseudopotentials and the projector augmented wave method:

The core electrons usually have higher kinetic energy and so need a bigger

number of plane waves to represent them, while the chemical bonds are mostly

associated with so-called valence electrons, far away from the core. The practical way

is to simplify the plane waves for the core electrons since they play only a slight role

in the bonds. This is the basic idea of the pseudopotential method. Generally, a KS

valence orbital is smooth in the bonding region and oscillates rapidly near the core;

the pseudopotential method replaces the oscillating part of the valence orbital with a

smoother pseudo-orbital , associated with a weaker pseudopotential

in the core, while keeping them the same as the true all-electron potential and orbital

outside the core.

There are different ways to construct the pseudopotential. The first one is

norm-conserving pseudopotentials which try to make the pseudo-charge inside the

core the same as the all-electron charge there. However, it is hard to generate this kind

pseudopotential with both transferability (one pseudopotential can be used in different

environment) and softness (one only needs a few plane waves). An alternative method

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37

called the ultrasoft pseudopotentials relaxes these constraints on the valence

pseudo-orbitals and can achieve better accuracy than norm-conserving

pseudopotentials for the same small number of plane waves.

FIG. 2-5: Schematic illustration of a pseudopotential orbital and the corresponding

all-electron orbital.

In VASP, the projector augmented wave method is used, which is a

combination of the ultrasoft pseudopotentials and linear augmented plane wave

methods that partition space into the space around each atom and an interstitial region.

Then the orbital inside the sphere can be represented by spherical harmonic functions

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38

and outside by an expansion in plane waves. The PAW method is quite

computationally efficient and can be as accurate as the all-electron method.

The plane-wave basis method can also be applied to atoms or molecules by

building a large super cell around the molecule to minimize the influence from its

periodic images. However, the LCAO method is better suited to these situations.

2.4.2 Gaussian and Gaussian Type Orbitals

In the localized atomic orbital method, the orbitals are written as the sum of

atomic-like orbitals

[2-31]

So the problem is to build a proper basis and to optimize the coefficients.

In the 1930s, the Slater-type orbitals were introduced to describe the atomic orbitals,

where N is a normalization constant; a,b,c control angular momentum, and

controls the width of the orbital. STOs can describe the shape of an atomic orbital

quite closely, but do not permit analytic integrations for molecules. Thus STOs are

generally only used for atomic problems and most molecular problem employs GTOs.

A Gaussian type orbital can be expressed as:

GTOs are not atomic-orbital-like, because they decay too fast with the increase of r,

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but they are much easier to use since the product of two Gaussian functions centered

on two different atoms is a finite sum of Gaussians centered on a point along the axis

connecting them. The solution for gaining both accuracy and efficiency is to first use

GTOs to represent STOs, then to use the resulting STOs to describe the atomic

orbitals. For example, the minimal basis set for an atomic orbital is called STO-3G,

which means that three GTOs are used to define one STO and one STO is used to

define an AO.

FIG. 2-6: Shape of Slater type and Gaussian type functions.

Another way, called Pople basis set, splits the core and valance electrons and uses

different numbers of GTOs to describe them.

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40

FIG. 2-7: One sample notation for the Pople basis set used in Gaussian

Gaussian[53] is a computational code using the Gaussian-type orbitals (GTOs)

to expand the atomic orbitals. It is widely used in chemistry and molecular physics. In

theory, the bigger the basis set used in the Gaussian code, the better the result we can

get. In practice, accuracy still depends on the real situation: a large basis set without

diffuse function may be worse in anions and for some big molecules, and it may be

expensive and unnecessary to apply a large basis set. DFT is less dependent on the

size of the basis set than are the wavefunction methods. However, a convergence test

to get a suitable basis set for the system is still needed.

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Chapter III

Ice phases under ambient and high pressure:

Insights for density functional theory

3.1 Introduction

3.1.1 Interactions inside solid ice

As mentioned early in the first chapter, water and its related systems play a

crucial rule in nature and human life, and attract lots of interesting from scientists.

Although the structure of liquid water has been experimentally researched for a long

time, the theoretical understanding is still poor and under debate[54], partly due to the

difficulty to describe accurately the interactions inside it. Consequently, we start with

the relatively easier water structure: ice, aiming to understand the basic relationship of

hydrogen bond and van der Waals interactions between water molecules.

Ice is the solid state of water. It exhibits a complex phase diagram [See Fig. 3-1]

with a hexagonal ice-Ih (also known as ice one) structure under ambient conditions.

Ice-Ih can exist down to 73K temperature and up to 0.2 Gpa pressure, as a result,

Ice-Ih is the most common ice phase on the Earth's surface. In fact, almost all the

nature snow, ice is in this structure. Because of its strong relevance to human

activities such as existence of life and regulation of global climate, ice-Ih has been

widely studied both experimentally and theoretically for hundreds years. Ice-Ih can

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42

transit into other phases with the changing of temperature and pressure. The rich

phase diagram is mostly coming from the relative intermolecular interactions of

hydrogen bond and van der Waals dispersion forces.

Fig. 3-1 Observed and predicted water ice phase diagram[55]. One atmospheric pressure=101kPa

=1.01x10-4

GPa.

The water molecule is made of two hydrogen atoms and one oxygen atom

linked by strong covalent bonds, while ice is a molecular crystal with intermolecular

interaction arising from two non-covalent bonds: hydrogen bonds and relatively

weaker but important van der Waals (vdW) interactions.

Generally speaking, a hydrogen bond is the electromagnetic interaction between

electronegative atoms, such as nitrogen (N), oxygen (O), fluorine (F), and

electropositive hydrogen atom. Hydrogen bond mostly happens between molecules

with strength less than 10 kcal/mol, (although made with F may be up to 30 kcal/mol),

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much weaker than covalent bond or metallic bonds. The most common form of

hydrogen bond is found between the water molecules where each oxygen atom can

form two bonds with hydrogen atoms from other two water molecules. Hydrogen

bonds strongly affect the liquid and crystal structures of water, for example, they help

to create the hexagonal structure for ice-Ih with 109.5。angle between two hydrogen

bonds, which is quite close to tetrahedral angle. At the other hand, the van der Waals

interaction is a long range attraction between molecules arising from a force between

the instantaneous induced dipoles. The vdW interaction is much weaker than normal

chemical bonds, but plays an important role in properties of many materials and

defines many properties of organic structure, which we will see next chapter.

Especially, the vdW interaction can have significant influence on the structure when

the intermolecular bond in ice is the relative weak hydrogen bond. Particularly, when

under high pressure, hydrogen bond strengths decrease significantly because the

nearest-neighbor water-water distances increase in comparison with those of ice-Ih

and the hydrogen bonds twist due to configuration distortions[56]. At the same time,

vdW interactions increase because layers of water molecule are packed much closer.

Thus the vdW interactions become more important in determining the properties of

ice structures at higher pressure[14]. This chapter follows closely Ref [57].

3.1.2 Density functional theory for solid ice

As a result, an accurate description of its properties requires the proper treatment of

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these intermolecular interactions[14], over a range of pressures and temperatures, as

can be seen in the phase diagram[55]. Many theoretical methods can be used for those

weak interactions, such as CCSD mentioned in chapter 2, they can be highly accurate

but with expensive calculation and so impossible to apply to bulk solids at present.

Density functional theory can be a good substitute for them. DFT is exact in theory; in

practice some approximations must be made. These approximations often yield

successful result but still have problem in same situation. One of them is that the

conventional functionals mostly fail for the weak bonds. However, recent years, with

the development of new density functionals and vdW dispersion corrections, density

functional theory has become a method of choice for this class of problems.

As discussed in chapter two, in the density functional theory, while the exact

general form of remains unknown, many exact conditions on have been

discovered. Density functional approximations can be developed to satisfy these

known conditions, or to fit data sets, or both. Many authors have employed DFT with

the generalized gradient approximation (GGA)[8,12] to study the properties of liquid

water and ice. Popular GGA functionals are less useful where vdW interactions are

important[58], and this can explain why certain GGAs underestimate the density of

liquid water[58] (PBE by 15%, revPBE[59] by 30%) and the sublimation energy for

high-pressure ice[14] (PBE by 15% for ice VIII) compared to experiment. As a result,

new functionals which more accurately describe both hydrogen bonds and vdW

interactions are strongly needed. The meta-GGA[60] is a natural way to improve

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accuracy further by making use of additional semilocal information (e.g., the

Laplacian of the density and/or the kinetic energy density ). Previous tests

showed that meta-GGAs yield better results than GGAs for covalent, ionic, and

metallic solids, including atomization energy, lattice constant, bulk modulus, and

cohesive energy[13]. Recently, it was found that a meta-generalized gradient

(meta-GGA) approximation to can describe the hydrogen bond[61] and vdW

interactions very well[36]. Motivated by this observation, we apply several

meta-GGA[34] functionals to study the lattice mismatch problem[29] of ice-Ih with

β-AgI, the sublimation energy in three different phases of ice, and the structural phase

transition pressures of ice.

Because the standard semilocal density functionals are unable to describe

the long-range part of the van der Waals interaction, several long-range vdW

corrections have been invented. In our work shown here, we apply two of them, the

DFT+D2[47] and vdW-DF[48] methods, to compare with our meta-GGA results.

Generally, in this chapter, we study the performance of GGA and new

meta-GGA functionals (including TPSS[32], revTPSS[13], and MGGA_MS[35]) on

geometry, sublimation energy, and transition pressures of various ice phases at

absolute-zero temperature, without zero-point vibration effects. Furthermore, by

comparing these results with the ones from density functionals corrected for

long-range vdW interaction (such as TPSS+D2[47], optB88-vdW[62] and

optB86b-vdW[62]), we aim to understand the advantages and limitations of all these

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46

approximations and the extent to which they account for the contributions of

hydrogen bonding and van der Waals interaction. As we will see later, meta-GGA

functionals reduce the lattice mismatch value of ice-Ih and β-AgI from 6% (PBE) to

around 3%, reasonably close to experiment. Additional calculations for high-pressure

ice phases indicate that the older meta-GGAs TPSS and revTPSS still have trouble

with vdW forces while the newer MGGA_MS is able to describe these interactions

well. However, by adding the appropriate long-range vdW corrections for solids,

revTPSS also tends to work well for the high-pressure phases.

3.2 Computational details

The Vienna Ab initio Simulation Package (VASP)[51,52] in version 5.2.12

has been used for the DFT calculation. VASP is a plane-wave code within the

projector augmented wave (PAW) method (details in chapter two). The “hardest”

PAW potentials available for H and O atoms were used for the sake of high accuracy

in the presence of short O-H bonds[29]. Ice-Ih was modeled using Bernal-Fowler’s

proton-ordered, twelve-water-molecule periodic model[28]. Ice-II and ice-VIII

structures were obtained from experiment: twelve molecules in a trigonal cell for

ice-II[63] and eight molecules in a tetragonal cell for ice-VIII[64]. The energy of the

isolated water molecule was calculated within a 10×11×12 Å3

box.

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47

FIG. 3-2: unit cell of three different ice phase, oxygen atoms are shown in red and

hydrogen atoms are shown in white

In a convergence test with the PBE and TPSS functional for ice-Ih, the total energies

were computed with the kinetic energy cutoff increasing from 900 eV to 1400 eV and

the Brillouin zone k-mesh from 2×2×1 to 4×4×4.

FIG. 3-3: (a) Convergence of the total energy with respect to energy cutoff used for

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48

revTPSS for all three ice phases (b) convergence of the sublimation energy with

respect to energy cutoff (c) convergence of total energy with respect to k-mesh.

From FIG 3-3(b), we can see that energy cutoff larger than 700eV is enough to make

the sublimation energy change within 1meV/H2O, while the total energy FIG. 3-3 (a)

needs more than 1100 eV to make the change within 5 meV/H2O to get a precise

equilibrium volume; on the other hand, the energy is less sensitive to the k-mesh,

larger than 2x2x2 is sufficient to make a revTPSS calculation result fully converge for

all three phases. Based on these tests, the optimizations for three ice crystal

geometries and total energies were performed for each functional using a plane wave

basis with a kinetic energy cutoff of 1200 eV and a 4×4×2 k-mesh to ensure

convergence.

All our calculations were self-consistent. The c/a lattice-constant ratios were set to

their experimental values, since the supplemental information of Ref. [ ] says that

optimizing these ratios affected the volume per molecule negligibly (by less than 0.02

Å3/H2O for ice-VIII).

3.3 Results and Discussion

3.3.1 Lattice mismatch challenge for ice-Ih and β-AgI

When two materials with different lattice constants are brought together by

deposition of one material on another, the lattice mismatch is defined as

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49

[3-1]

where a1 and a2 are the lattice constants for the two separated materials. Lattice

mismatch is a critical parameter for thin film growth on a crystal. For example, large

lattice mismatch will prevent the growth of a defect-free epitaxial film unless the

thickness of the film is below a certain critical thickness[65]. Consequently, good

prediction for lattice mismatch is important for theoretical simulation of such

phenomena.

One application of this is the cloud seeding with β-AgI[66]. In a cloud, an ice

crystal seed can make the crystallization process hundreds times faster than in super

cooled water. In cloud seeding, the crystalline β-AgI can be used to produce artificial

rainfall, because β-AgI smoke provides seed crystals used as the artificial ice nuclei in

clouds for rain-inducing ice crystallization. This application is based on the fact that

the mismatch between the lattice constants of ice-Ih and crystalline β-AgI is only

about 1% at 273 K (~2.2%[29] after extrapolating to low temperature at 10-30 K).

However, Feibelman[29] in his early work pointed out that the lattice constants of

ice-Ih and β-AgI as predicted by LSDA[25] and some GGA-level density functionals

produce a significantly too-large mismatch value (~8% for LSDA,~6% for PBE)

compared to experiment. This so-called lattice mismatch puzzle led to doubts about

using DFT approximations for water-material interactions. Ice-Ih has a hexagonal

crystal structure, in which hydrogen bonds constitute up to 90% of the whole

interaction[14]. On the other hand, β-AgI is a solid with strong van der Waals

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50

interactions because of the heavy I ions. Therefore, to get an accurate mismatch

value, the functional should be able to describe both the hydrogen bond and the

dispersion interaction simultaneously, a challenge for most semilocal density

functionals.

The first four rows of Fig. 3-4 show the relative errors for lattice constants

and also the lattice mismatch computed from four widely-used GGA functionals.

Among them, PBE gives relatively small errors for both structures, but, because of the

opposite directions of the errors, the mismatch value calculated by PBE is too large.

Conversely, revPBE finds the smallest mismatch among GGAs, but overestimates the

lattice constants for both solids too much. It is known that PBE overestimates

hydrogen bonding and fails for the vdW interactions. This explains why PBE

overbinds ice-Ih while underbinds β-AgI.

We now discuss the results computed by meta-GGA functionals, the older

TPSS and revTPSS and the newer MGGA_MS. We also present the results of three

functionals with long-range vdW corrections: TPSS+D2[47], optB88-vdW[62] and

optB86b-vdW[62]. From Fig. 3-4, we can see that, except for TPSS which performs

similarly to PBE, the other two meta-GGAs show good agreement with experiment:

The relative errors for ice lattice constants are smaller than 1% while the lattice

mismatches are around 3% compared to the experimental value of 2.3%. This

indicates that these meta-GGAs have the potential to better describe the ambient ice

structure.

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Fig. 3-4: Percentage error of lattice constant for the ice-Ih on β-AgI lattice mismatch problem.

Experimental lattice constants and mismatch value are taken from Ref. [28 ].

On the other hand, all functionals with long-range vdW correction give as large a

mismatch value as the GGAs do. Fig. 3-4 shows that, although these functionals give

accurate results for β-AgI, they underestimate the lattice constants of ice-Ih more than

2%, worse than PBE. This shows that, by including the long-range vdW corrections in

these density functionals, one achieves more accurate results for solids with strong

vdW interactions, but less accurate results in ice structures with hydrogen bonds.

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Based on these results, the meta-GGA functionals revTPSS and MGGA_MS

show the best performance overall. However, as vdW interactions play only a minor

role in ice-Ih, studies of properties for different ice phases are needed to understand

the ability of various functionals to describe hydrogen bonds and van der Waals

interactions in ice.

Table 3-1: Lattice constant of ice-Ih and β-AgI, lattice mismatch value for different

density functionals

Functionals ice-Ih lattice constant

(Angstrom)

β-AgI lattice constant

(Angstrom)

Mismatch

(%)

LSDA 4.16 4.50 7.85

PW91 4.39 4.68 6.24

PBEsol 4.28 4.56 6.17

PBE 4.43 4.67 5.36

RPBE 4.6 4.80 4.09

TPSS 4.45 4.69 5.22

revTPSS 4.50 4.64 3.05

MGGA_MS 4.48 4.63 3.27

MGGA_MS2 4.49 4.63 3.07

TPSS+D2 4.37 4.58 4.77

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optB88-vdW 4.41 4.65 5.25

optB86b-vdW 4.39 4.7 4.99

optPBE-vdW 4.48 4.61 4.89

Expt. value 4.497 4.594 2.17

3.3.2 Sublimation energy for ice phases under ambient and high

pressure

As mentioned before, solid ice exhibits a rich and complex phase diagram. We

next report our tests on ice-Ih at ambient pressure, and on two other proton-ordered

phases, in order of increasing pressure: ice-II[63] and ice-VIII[64] (shown in Fig. 3-1)

using GGAs, meta-GGAs, and semilocal functionals with vdW correction. We also

compare our computed results with experiment. Previous work[14] indicates that, for

the phases at higher pressures, hydrogen bond strengths decrease significantly. At the

same time, vdW interactions increase because layers of water molecule are packed

much closer. Thus the vdW interactions play a more crucial role in determining the

properties of ice structures at higher pressure.

Next we discuss the sublimation energies for these three ice phases with different

functionals. The sublimation energy is defined as the difference between the energy of

an isolated water molecule and the energy per water molecule in the solid structure. It

represents the energy change from the solid to the gas phase, including all the

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intermolecular interactions in the solid structure. For ice structures, intermolecular

interactions consist of hydrogen bonds and vdW dispersion forces. By comparing the

results for various ice phases, we can analyze the performance of DFT for these two

weak interactions. Table 3-2 shows the computed results of sublimation energies from

various functionals. The total-energy difference

with respect to the ice-Ih phase is calculated for high-pressure phases

and the results are shown in parentheses. Experiment shows that ice-II is almost as

stable as ice-Ih with only about 1 meV/molecule, while ice-VIII is less stable

than ice-Ih by 33 meV/molecule. From Table 3-2, PBE slightly overestimates the

sublimation energy for ice-Ih, while underestimating it by 40 meV/molecule for ice-II

and by 117 meV/molecule for ice-VIII. The main reason has been explained by

previous work[14]. TPSS and revTPSS yield acceptable sublimation energy for ice-Ih,

but still fail for high-pressure phases. The revTPSS total energy difference between

ice-VIII and ice-Ih is about 5 times larger than experiment due to underestimation of

the sublimation energy for high pressure. Since vdW interactions become stronger

with increasing pressure, this indicates that these semilocal functionals do not

describe vdW well. Therefore, we also show the sublimation energy of the TPSS+D2

method, which is the TPSS meta-GGA with long-range vdW correction. From Table

3-2, we notice that, with the vdW correction, total energy differences ( are

greatly improved compared to GGAs, giving values much closer to experiment.

However, we still notice that the sublimation energies predicted by TPSS+D2 are

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significantly too large, with mean absolute relative error up to 16%. The strong

overbinding can lead to a poor description for structural properties like volume. As we

already found in part I and also in Fig.3-5 (a), TPSS+D2 underestimates the

equilibrium volume for all three phases. The DFT+D2 method is constructed for

molecules and clusters, and seems to over-count the vdW interactions in solid. This

phenomenon can also be found within the vdW-DF method, as we can see for

optB88-vdW in Table 3-2.

For lack of an accurate vdW correction for solid ice, we employed the vdW

data from Ref. [68] which adds the influence of vdW interactions within the scheme

of Tkatchenko and Scheffler[67], as calculated to correct the PBE0 hybrid functional.

Since this method has been shown to be largely independent of the employed DFT

approximation and works well for solids[67][68], we add this correction to our

meta-GGA revTPSS result, and find that the total energy yields precise sublimation

energies (with MARE of 3.5%) and a significant improvement for . However, for

the super high-pressure phase ice-VIII, this revTPSS+vdW predicts an energy

difference U (~75 meV) slightly worse than optB88-vdW (26 meV), in comparison

to experiment (33 meV), but still performs better than the PBE GGA (177 meV).

Finally, the new MGGA_MS works well and performs quite similarly to the

revTPSS+vdW with 4.3% MARE and with 70 meV for ice-VIII. MGGA_MS

has no long-range vdW correction, and seems to capture part of the vdW interaction

in ice by itself.

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Table 3-2: Sublimation energies of ice-Ih, -II, -VIII (omitting zero-point energy effects). The

total-energy differences compared to ice-Ih (in

parentheses) and the mean absolute relative error of the sublimation energy averaged over the

three phases.

a: Experimental values are taken from Ref. [ 68 ], b: with zero-point energy contribution

removed.

Ice-Ih Ice-II Ice-VIII MARE (%)

LSDA 943 896(47) 813(130) 47.5

PBE 636 567(69) 459(177) 10.5

TPSS 587 502(85) 380(207) 18.49

revTPSS 570 507(63) 423(147) 16.67

MGGA-MS 602 586(16) 532(70) 4.3

MGGA-MS2 584 553(31) 505(79) 8.65

PBE0a 598 543(55) 450(148) 11.61

PBE0+vdWa 672 666(6) 596(76) 7.61

revTPSS+vdW 644 630(14) 569(75) 3.5

TPSS+D2 720 690(30) 675(45) 16.11

OptB88-vdW 696 699(-3) 670(26) 15

OptB86b-vdW 706 701(5) 666(40) 15.42

DMCa 605 609(-4) 575(30) 0.39

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Expt.a,b

610 609(1) 577(33)

3.3.3 Transition pressure between ice phases

To better understand how functionals perform for van der Waals interactions,

we go on to study the ice phase transitions under pressure, computing the phase

transition pressure between ice-Ih and ice-II or ice-VIII. Fig.3-5 (c) shows the energy

versus volume curve for ice-Ih and -VIII phases for MGGA_MS, fitted by the

Birch-Murnaghan equation of state[69] :

[3-2]

where are the equilibrium volume, total energy, and bulk modulus,

and B '

0 is the pressure derivative of the bulk modulus.

The equilibrium volume, lattice energy, and bulk modulus for each phase are

also obtained from the same EOS parametric fitting, and the results for volume and

energy are illustrated in Fig. 3-5 (a) & (b). Then the transition pressure can be

obtained by constructing the common tangent line (dotted line in Fig. 3-5 (c)) for the

two EOS-fitted energy-volume curves.

We apply this approach with the tested functionals to get the transition

pressures from ice-Ih to -II or -VIII, and the results are given in Fig.3-5 (d). The

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horizontal axis shows the transition pressure Ptr for Ih to II, and the vertical axis

shows Ptr for Ih to VIII. Because the ice-Ih and -II phases are almost equally stable,

the transition pressure is quite small (~0.02 GPa), and only optB88-vdW and

MGGA_MS give reasonable predictions. All other functionals predict a transition

pressure larger than experiment. For ice-VIII, where the experimental value is 0.44

GPa, optB88-vdW still gives the best result while other functionals with vdW

corrections and MGGA_MS also work well. From Fig. 3-5 (d) for the transition

pressure Ptr, and from Fig. 3-5 (b) for the energy difference , we can see a grouping:

vdW-corrected functionals cluster in a close range around experiment, while GGAs,

TPSS and revTPSS fall farther away from this range. Clearly, adding the vdW

correction contributes to the improvement of transition properties. Also, notice that

the MGGA_MS results fall in the close range with these vdW-corrected functionals.

This indicates that MGGA_MS captures at least part of the vdW interactions in ice.

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Fig. 3-5: (a) Relative lattice volume (△V0) of the high-pressure ice-II and -VIII with respect to

the lattice volume of ice-Ih and (b) relative total energy (△U0) (c) The energy versus volume

curves of the ice-Ih and ice-VIII systems with MGGA_MS. The dotted line is the common tangent

line obtained from the Birch-Murnaghan EOS. The slope of the straight line gives the transition

pressure (Ptr). (d) Transition pressures (Ptr) from ice-Ih to the phases ice-II and -VIII.

PBE0+vdW results and experimental values are taken from Ref. [68 ]. The calculated values do

not include zero-point vibration effects. These effects are removed from the experimental

total-energy changes, but not apparently from the experimental equilibrium volumes. Ref. [ 68]

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Table 3-3: Equilibrium volume of ice-Ih,-II and –VIII, transition pressure from ice-Ih

to –II and –VIII

Ice-Ih

volume

(Å3)

Ice-II

volume

(Å3)

Ice-VIII

volume

(Å3)

Ptr Ice-Ih to

–II

(Gpa)

Ptr Ice-Ih to

–VIII

(Gpa)

LSDA 27.47 21.73 17.57 1.39 2.26

PBE 30.75 24.83 20.62 2.04 3.18

TPSS 30.85 25.03 20.38 2.30 3.84

revTPSS 32.05 25.39 19.62 1.57 2.7

MGGA_MS 31.83 24.82 19.81 0.02 0.82

MGGA_MS2 32.20 25.00 19.58 0.72 1.05

PBE0a 30.98 24.84 20.25 1.45 2.22

PBE0+vdWa 29.88 23.63 19.69 0.16 1.19

optB88-vdW 30.23 23.68 18.98 -0.05 0.37

optB86b-vdW 29.78 23.39 18.74 0.12 1.42

TPSS+D2 29.33 22.41 18.95 0.74 0.61

DMCa 31.69 24.70 19.46 -0.09

Expt.a 32.05 24.77 20.09 0.02 0.44

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3.4 Liquid water

Experimental liquid water structure is measured by x-ray and neutron

diffraction experiments, while there are difficulties in interpreting these data

theoretically. As a result, till now there are still considerable controversies. The

structure of liquid water gained from molecular dynamics simulations using

DFT-PBE, revPBE and BLYP, did not agree well with experiment as shown earlier.

These reflect the needs of more accurate functionals which can better describe

hydrogen-bonds and vdW inside the liquid water and our work suggests the possible

way to achieve this. However, it is noticed that a meaningful and practical AIMD

simulation for liquid water need at least a 64 water molecules systems with a 20 ps

simulation process which is impossible to run for high-level functionals now without

technological and algorithmic breakthrough. Due to this, the problem for liquid still

exists and need to be answered in the future.

3.5 Conclusions

In summary, we have studied hydrogen bond and van der Waals interactions

within various ice structures using different density functionals. First we found that

two meta-GGA's, revTPSS and MGGA_MS, essentially solve the GGA lattice

mismatch puzzle[29] for ice-Ih on β-AgI, and we argued that only a functional like

MGGA_MS, that reliably describes intermediate-range van der Waals interaction as

well as the hydrogen bond, can reliably solve this kind of problem.

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Then we found that meta-GGA yields a better description than GGA for the

sublimation energy and equilibrium volume of low-pressure ice phases, a difficult

problem for semilocal functionals. In particular, meta-GGA can describe these

properties at least as accurately as vdW-corrected GGAs can, even without relying on

vdW dispersion corrections, as demonstrated for MGGA-MS: The results (especially

the volume per molecule) for ice-Ih, –II, and -VIII are in quite good agreement with

experiment, while the sublimation energy for ice-VIII is slightly underestimated but

still improved over GGAs.

We find that MGGA_MS is an accurate method for computing ambient and

high-pressure phases of ice, although it needs a long-range vdW correction under

super-high pressure. We have argued elsewhere[36] that MGGA_MS has the right

dimensionless ingredients to recognize covalent, metallic, and weak bonds. Other

meta-GGAs can be built from these dimensionless ingredients[70], and long-range

vdW[71,72] corrections of lesser importance can be constructed for them. Such

functionals may be useful for many problems, including the problems of liquid water

and of DNA/RNA[36].

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Chapter IV

What forces twist the Deoxyribonucleic Acid?

Interactions between nucleic bases with density

functional theory

4.1 Introduction

4.1.1 Deoxyribonucleic acid structure and nucleic bases

In biology, nucleic acids are large biological molecules which function in

encoding, transmitting and expressing genetic information for almost all forms of life

as well as some nonliving entities on the earth. "Nucleic acid" is the overall name for

the DNA (deoxyribonucleic acid) and RNA (ribonucleic acid). Mostly, DNA

molecules are double-stranded while RNA molecules are single-stranded, and DNA is

the basic genetic instructions for most living organisms except viruses, which use

RNA. As a result, DNA attracts lots of interesting from different scientific fields

including biology, chemistry and physics. Since the first isolation of DNA by a Swiss

physician in 1869[73], a lot of efforts were made to find the structure of DNA

molecules. Based on the X-ray diffraction patterns which showed that DNA had a

regular structure[74], in 1953, James Watson and Francis Crick suggested the

double-helix model[75] of DNA structure which was supported by latter experimental

evidence[76]. Nowadays, this so called double-helix twist DNA model may be the

most well-known biology molecule that can be seen everywhere, even on the cover of

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TIME and Waston, Crick and Wilkins jointly received the Nobel Prize in Physiology

or Medicine in 1962.

FIG. 4-1:The structure of the DNA double helix[77]

As we can see from FIG. 4-1, the DNA molecule is made of two long chains.

Each coils around the center axis, molecule called nucleic bases link to the chains.

The basic unit of DNA is nucleotides which build the DNA by repeating itself through

the space. The sugar and phosphate group in the nucleotide alternatively link with

each other to make the chains (called the backbone of DNA) and nucleic bases are

linked to the sugar.[78] One nucleic base connects with another on different

backbones by hydrogen bonds. With this force the two chains are linked together. One

DNA molecule can contain millions of nucleotides, which means millions of nucleic

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bases[79]. However, there are only four kinds of nucleic bases ,classified into two

types: the purines, A (adenine), G (guanine); the pyrimidines, C (cytosine), T

(thymine) and they are linked following the pair rules; A only pairs with T, G only

pairs with C. In sequences of four possible nucleic base pairs: A:T, T:A, G:C, C:G,

DNA can be well suitable to the storage of enough encoded genetic information to

make every human being different. .

FIG. 4-2: Nucleic bases, A (adenine), G (guanine), C (cytosine), T (thymine)

(source from websit)

4.1. 2 Twist of DNA and DNA base pair steps

The most famous and important structure properties of DNA are its two backbones,

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which run in opposite direction to each other and therefore anti-parallel twist together.

As shown in FIG. 4-3, the B-DNA, which is the most common one found in living

animals, has an average 10 pairs per turn of length 34 Å long, so that is ~36 degree

twist and 3.4 Å vertical distance between every two neighbor nucleic base pairs[80].

The large size of DNA molecule makes it difficult to search for the inside

interactions that build this structure both experimentally and theoretically. However,

with the limited sorts of nuclei bases, it is quite practical and reasonable to understand

the physical properties of stacked nucleic base pairs first, then by repeating this small

part sequentially in the space, a better understanding of the whole structure can be

gained. Moreover, the replicating of DNA and binding interactions with other organic

molecules such as proteins generally happen at the base pair level. As a result, to

interpret the interactions between these nucleic bases is quite important to explain the

microscopic as well as the macroscopic properties of DNA. To do this, the base pair

configuration needs to be studied and precisely described.

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FIG. 4-3 Model of the double helix of B-DNA with 10-10.5 base pairs per turn

Based on the X-ray experimental result, because of the coil of backbones,

geometry displacements may happen between the neighbor pairs, generally, there are

six sorts of displacements called base pair steps which can exist: shift, tilt, slide, roll,

rise, twist, which are shown in FIG. 4-4. For B-DNA, the steps of shift, tilt, slide, roll

are zero or small, with the rise around 3.4 Å and twist around 36 degree. As a result in

my work the influences of the earlier four steps are ignored and only the structural

steps parameters of rise and twist are studied.

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FIG. 4-4: Six possible nucleic base pair steps[81]

It is commonly accepted that there are two sources responsible for the base pair

geometry structure[82]. One is the force from the backbones: the

hydrophobic-hydrophilic interactions with the sugar-phosphate group and solvent

molecule （mostly water）[83,84]. Another is the interactions between the bases which

include the hydrogen bond[85] linking the two adjacent bases from the opposite

backbones and the pi-pi stacking interaction[86] between stacked based pairs.

However, the relative contribution of each is still unclear[82]. In experiment, it is hard

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to distinguish and separately research on each force; in theory, for the reason of the

complex structure of backbones and liquid solvent it is difficult to impose the

influence of the sugar-phosphate backbones on the nucleic base pairs. Traditionally,

the first principle theoretical calculations mostly apply to the base pairs without the

backbones.

4.1.3 Ab initio methods for nucleic bases

Since the noncovalent weak interactions between DNA and other

bio-molecules such as RNA and proteins are important, they play a crucial role in the

transcription and replication of nucleic acid, interaction of DNA with proteins and

RNA, and intercalation of chemotherapy drugs to DNA. As the result of this, the

ability to describe these interactions precisely is a basic requirement for theoretical

models which can be use for understanding gene mutation, drug design and so on.

Wave function based methods such as MP2, CCSD are widely used in

quantum chemistry calculations and these methods often can give relatively reliable

results for the interactions between molecules, especially these weak ones.[87] Their

accuracy relies on large basis set which leads to huge computational demands, (O(N5)

for MP2, O(N6-7

) for CCSD(T)). In this sense, their application is limited. The high

level CCSD(T) using triple-zeta basis set, which can work as the benchmark for other

ab initio methods, only become available in recent years for small size of molecule

system[88].

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On the other hand, density functional theory (DFT) is an efficient and often

accurate-enough way which scales well with system size. The GGA PBE and

hydrid-B3LYP also are widely used for studying many chemical systems with tens or

even hundreds of atoms. However, the application of these conventional density

fucntionals is limited due to the poor results[82] of these functionals to describe

systems in which the van der Waals dispersion forces are significant. In recent years,

several new approximations are designed for improving existing density functionals to

deal with this dispersion problem, as we introduced in chapter two, and some of them

like vdW-DF[48] are well tested in many systems like benzene dimer, grapheme

layers, stacked nucleic bases and base pairs[89]. These tests suggest that the

dispersion force plays an important role in determining the base pair steps of rise and

twist. Based on the result we got in the last chapter, the performance of these

meta-GGA functionals (especially the newly developed MGGA_MS) needs to be

analyze for this system with a combination of different weak interactions including

hydrogen bond, dispersion force, as well as electrostatic interactions. Besides, the

DFT+D method, in which the dispersion interactions are added by empirical pair

interaction term, and M06-L fucntionals, which fitted its empirical parameters to

databases that includes weak-interaction dimers, have shown promise for weak

noncovalent interactions, but have not yet been fully applied to these biologic

problem, unlike the vdW-DF method. As a result, in the following, we analyze the

performance of early mentioned meta-GGA fucntionals along with M06-L and

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DFT+D2 for nucleic base pair steps.

In the following part of this chapter, the presence of a colon ( : ) represents the

hydrogen bond interactions between nucleic bases, a hyphen ( - ) represents the

stacking interactions.

4.2 Computational details

All DFT calculations are performed using the Gaussian program[53];

Gaussian is a computer package for chemical calculation using Gaussian orbitals

(more details introduced in chapter two). All GGA level functionals and most

meta-GGA functionals calculations are performed with Gaussian 03, while the M06-L

functional is only available above the version Gaussian 09. I have ensured that the

result obtained by Gaussian 03 and Gaussian 09 for the same functionals are exactly

same as shown in FIG. 4-5. The basis set 6-311++G** is used for all calculation based

on the energy convergence test for C-C stacking pair.

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FIG. 4-5: Total energy of Guanine dimer with rise from 3.4 Å to 4.0 Å

calculated with GGA PBE using Gaussian 03 and Gaussian 09

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FIG. 4-6 Convergence tests of total energy with respect to basis set used for

the GGA PBE calculation for (a) Cytosine nucleic base (b) stacked Cytosine dimer

The initial structure of A (adenine), G (guanine), C (cytosine), T (thymine)

and A:T, G:C pairs are following the nomenclature and coordinate system of Olson et

al., used in the Nucleic Acid Database[90]. The equilibrium structure for each

functional of individual nucleic bases and pairs are fully optimized using each

functionals except for the DFT+D2 method, for the lack of SCF process for it, in

which the van der Waals corrections are calculated based on the optimized DFT

structure. For example, the vdW correction energy of PBE+D2 is computed using the

PBE structure.

4.3 Results and Discussion

As discussed earlier, the geometric and energetic properties of nucleic bases

are determined by the hydrogen bond as well as the pi-pi stacking interaction. The

theoretical investigation makes it possible to isolate and study the influence and

relative contribution of each interaction. In this case, first of all, the hydrogen bonds

between WC pair A:T and G:C are researched, then the stacking interactions are

investigated with the dependence of the interaction energy on twist of undisplaced

stacked nucleic base dimers using various functionals. Furthermore, later in this

chapter, the interactions between stacked nucleic base pairs are studied to search for

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the reason for helical twist angle of DNA.

Since the experimental data from the X-ray crystallography are influenced by the

molecular environment, usually, there are discrepancies between theoretical prediction

of gas-phase structure and experimental values. Therefore, our DFT computed results

are compared with the mostly accurate wave function method which can be found in

literature.

Note that, even the influence of backbones is not including in the most

theoretical models, X-ray data shows that they also contribute in determining the

three-dimensional structure.[83,84]

4.3.1 Hydrogen bond interaction between Watson-Crick pairs

FIG. 4-7: Nucleic base pairs and hydrogen bonds (dot lines) between bases

The hydrogen bond interactions in DNA are dominated by the electrostatic

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contribution, the nature of which is discussed in chapter three. The network of

hydrogen bond inside the DNA is based on the relative interaction between nucleic

bases which build the WC pairs. As shown in FIG. 4-7, two hydrogen bonds occur

between the adenine and the thymine base pairs, and between the cytosine and the

guanine there are three hydrogen bonds. The double helical structure of DNA is due

largely to hydrogen bonding between these base pairs, which link one complementary

strand to the other, besides the role in determining the three dimensional structure, the

hydrogen bonds also contribute to the replication of DNA and adoption of protein and

base pairs because the weak strength of this bond makes it easy to break and reform.

As a result, these hydrogen bonds have been the subject of many

investigations both experimental and theoretical. Previous work shows that both the

high cost quantum chemistry methods and the DFT work successfully in the

description of the hydrogen bond inside DNA structure. The popular GGA-level DFT

can well describe this interaction. In my work, the meta-GGA level functionals

revTPSS, MGGA_MS and M06L are applied to the nucleic base pairs A:T and G:C to

check if this success can be maintained. Besides, it is believed that the van der Waals

dispersion force, even very weak, also contribute to the bond strength of base pairs,

hence, in this part, vdW corrections are also included using Grimme’s DFT+D2

methods to check the relative contribution of dispersion force.

4.3.1.1 Bond length and bonding energy for different density functionals

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In Table 4-1, the performances of various DFT fucntionals on bond parameters

within Watson-Crick base pairs A:T and G:C are given, together with the MP2

benchmark result. The bond length D is defined as the distance between the two

non-hydrogen atoms (see FIG. 4-7); the bonding energy Ebond is computed by

subtracting the total energy of individual bases from the total energy of the base pairs.

All energies are gained by full optimization for every structure.

Table 4-1 shows that, except for the LSDA functional other GGA and meta-GGA

functionals give relative close agreement with MP2 value for bond length and

bonding energy with different level. It is widely known that the LSDA functional

overestimates the hydrogen bond strength up to 50%, which can be proved with our

result: the bonding energy is significantly overestimated and the bond length is much

smaller than the MP2 value. Concerning the different performance of other

functionals for the bonding energy, it is the M06L that gives the best predictions with

the values of -14.24 kcal/mol and -26.87 kcal/mol for A:T and G:C, which is only

~0.8 kcal/mol smaller than MP2[89]. The PBE and MGGA_MS functionals work also

quite well with bonding energy 1-1.5 kcal/mol smaller, while revTPSS performs the

worst with ~3 kcal/mol error. However, all functionals give absolute error close or

smaller than chemical accuracy 1 kcal/mol per bond. As for the bond length, PBE

gives the best description of the geometry, while other functionals overestimate the

bond length by 0.03 Å-0.06 Å comparing to MP2 results but only have smaller than 1%

influence on 20 Å width of the base pairs.

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Table 4-1: Hydrogen bond length D and the bonding energy Ebond for the nucleic base

pairs A:T and G:C for different density functionals as long as MP2 method[89]

Pair Bond

Parameters

LSDA PBE MGGA_MS revTPSS M06L MP2

A:T DN1-N3 (Å) 2.64 2.83 2.86 2.89 2.87 2.83

DN6-O4 (Å) 2.73 2.89 2.94 2.92 2.91 2.86

Ebond

(kcal/mol)

-24.91 -13.90 -13.43 -12.37 -14.24 -15.1

G:C DO6-N4 (Å) 2.60 2.76 2.80 2.80 2.80 2.75

DN1-N3 (Å) 2.77 2.92 2.94 2.95 2.94 2.90

DN2-O2 (Å) 2.74 2.90 2.92 2.93 2.91 2.89

Ebond

(kcal/mol)

-40.30 -26.91 -26.33 -24.45 -26.87 -27.5

4.3.1.2 The contribution of vdW dispersion force for bonding between base pairs

Based on the result shown in Table 4-1, mostly, GGA and meta-GGA

functionals underestimate the bonding energy and overestimate the bond length; this

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indicates that there are at least some van der Waals dispersion interactions missing

from these functionals. To take into consideration and check the role of vdW, the

correction energies are added following Grimme’s DFT-D2 approach using pair wise

atom interactions. In FIG. 4-8, the mean binding energies per hydrogen bond for three

common used density functionals with and without the D2 correction are shown.

FIG 4-8: Mean binding energy for nucleic base pairs with density functionals

with and without vdW D2 corrections.

By adding the vdW dispersion correction, different functionals perform

variously. PBE+D2 method overestimates the mean binding energy by 0.9 kcal/mol,

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which is even larger than the magnitude underestimated by PBE itself. Form chapter

three, it is found that PBE usually overestimate the strength of hydrogen bond, this

can lead to the over counting after considering the vdW energy. At the other hand,

hybrid B3LYP and meta-GGA revTPSS perform well with the correction. The

absolute error decreases from ~1.1 kcal/mol to ~0.6 kcal/mol because of the better

description of hydrogen bond of these two. However, adding the vdW correction for

these functionals always leads some overestimation for the bonding energy.

4.3.2 pi-pi stacking interactions between stacked bases

Pi-pi stacking interactions are usually defined as the attractive interactions

between two parallel or face-to-face oriented aromatic systems. These interactions

play a fundamental role in many aspects of chemistry and biology[84,91]. For

instance, face-to-face pi-pi stacking interactions are responsible for the slippery feel

of graphite layers; pi-pi stacking in biology molecules is often important to the

structure and function of protein, RNA, DNA[92]. In fact, considering the huge

number of aromatic rings in the large biology molecules such as DNA, the individual

weak stacking effect can add up to powerful force that dominates the structure of

them. They are often believed to be the hand at work in the self-assemble[93] of

bio-molecules.

Though the importance of the pi-pi stacking interactions, the first model[91] of

them in which the electrostatics is thought to be major part, was proposed not until

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1990 and has been criticized by numerous research groups since then. Nowadays, it is

well accepted that the interactions are mainly determined by the interplay of

electrostatic effects, the van der Waals dispersion force, the exchange

(Pauli)-repulsion at short intermolecular distance and so on, however, there are still

los of debates on which one is dominates[94]. It would seem that the relative

contributions are highly dependent on the geometry of the molecules and the design

of the experiment and this phenomenon is still not well understood theoretically yet.

In a recent paper[95] of Grimme, he suggests that:

“The terms pi stacked or pi-pi stacking should merely be used as geometrical

(structural) descriptors but that specific (distinguished) attractive interactions

between pi electrons in such system do not exist.”

Density functional theory can accurately describe the electrostatic interactions

and the exchange repulsion, but those conventional GGAs, meta-GGAs cannot treat

the vdW part well, as shown in the former part of this thesis. Despite the know

shortcomings, it is still quite interesting to examine the performance of DFT on these

pi-pi stacking system, especially the new MGGA_MS, which is believed to capture

part of mid-range vdW interactions.

4.3.2.1 Benzene dimer stacking

The benzene (C6H6) dimer is the prototypical system for the study of pi-pi

stacking and has been quite extensively studied. The experimental binding energy is

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about 2-3 kcal/mol in gas phase[96]. This significantly weak binding energy makes

the benzene dimer difficult to study experimentally. It is also to be a challenging task

for theoretical methods to describe it accurately. Besides, it is the simplest example of

a pi-pi stacking system which can be found everywhere in biological molecules; as a

result, the high-level CCSD(T) method with very large basis set can be applied to

benzene dimer to get a quite accurate benchmark for other theoretical methods[97].

FIG. 4-8: Three possible dimer geometries and stacking energies[98]

FIG. 4-8 shows three possible configurations of benzene dimer: perfectly parallel

in sandwich shape (S), parallel displaced (PD), and edge-to-face or T-shape (T), and

their stacking energies with CCSD(T) at the complete basis set limit. These most

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accurate calculations prove the existence of this significantly weak binding energy.

Following, the ability of various density functionals for this system are investigated

based on S-shape (FIG. 4-8) structure.

FIG. 4-9: Stacking energies for S-shaped benzene dimer with 3.9 Å distance

calculated with various density functionals and MP2[99] compared to CCSD (T)

value (blue line).

The conventional functionals including GGA PBE, meta-GGA revTPSS, and

hybrid-B3LYP, are unable to give a proper description of the benzene dimer: they

predict a purely repulsive interaction, and as a result no binding position can be

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found. MGGA_MS improves this but can only capture small a part of the binding

energy with a large binding position at r=4.4 Å. M06L works well for this stacked

system with small energy difference and slight underestimation of bond position (3.8

Å), which is quite reasonable because the parameters of M06L are fitted to the

database (PPS5/05)[33] including the benzene dimer. The widely used quantum

chemical method MP2 overestimates the energy largely. In fact, based on previous

work, MP2 mostly tent o overestimate the pi-pi stacking energy[100]. The problem

of these density functionals indicates the need of new correction. Here I employed

the PBE+D2 vdW corrections and add it to the GGA PBE energy.

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FIG. 4-10: Binding curve (Etot) of benzene dimer obtained by adding DFT-PBE (EPBE)

energy to PBE-D2 dispersion correction (ED2-disp).

From FIG. 4-10, we can notice that for GGA PBE functionals by adding the

dispersion correction, even in a simple approach using only pairs of atoms, can greatly

improve the binding curve for the benzene dimer: the unbinding problem can be solved;

a global minimum at 3.8 Å with binding energy 1.8 kcal/mol can be gained, only

slightly larger than CCSD method. This indicates the importance of vdW interaction in

the pi-pi stacking at least for the benzene dimer system.

Because the symmetry of geometry and electrostatic potential of benzene, (see FIG.

4-11 (a)), there are no net dipole in benzene, existence of quadrupole-quadropole

interactions leads to the weak vdW force and these vdW and electrostatic (ES) forces do

not tend to change too much when one benzene is rotated above the other one. However,

in the nucleic bases, situation can be more complex because there is no symmetry of the

electron density distribution (see FIG. 4-11(b)), vdW and ES forces may change largely

due to the rotation of one base above another which lead to a much more complex

interplay of these forces.

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FIG. 4-11: Electrostatic potential surface for (a) benzene (b) adenine

4.3.2.2 Rise and potential energy surface (PES) for stacked nucleic bases

(i) Rise for stacked base pairs

Rise for nucleic bases is defined as the vertical distance between the mass centers

of two stacked antiparallel bases (see FIG. 4-12) because of the high repulsion

existence in parallel structures. To get the value of rise, the individual base structures

are optimized first with different functionals, then the intramolecular structure are

kept frozen in all the methods and only the intermolecular coordinates are varied to

find the minimum energy.

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FIG. 4-12: Rise defined as the vertical distance between stacked antiparallel C-C

nucleic bases

Rise for stacked bases are strongly determined by the bond interactions between

them, in TABLE, the rise for seven stacked pairs of A-A, C-C, G-G, T-T, A-G, A-C,

G-C are shown. The mean relative error (MRE) are gain by comparing to the rise

value of 3.4 Å for C-C, G-G, G-C and 3.3 Å for others, based on the previous work of

Sponer[101].

TABLE 4-2: Nucleic bases rise (Å) computed with several density functionals

and their MRE (Å) compared to Sponer work.

LSDA PBE revTPSS MGGA_MS M06L

A-A 3.28 4.08 4.30 3.71 3.48

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C-C 3.35 3.93 4.00 3.70 3.56

G-G 3.15 3.81 4.08 3.51 3.34

T-T 3.10 3.75 3.93 3.54 3.35

A-G 3.18 4.02 4.30 3.63 3.40

A-C 3.25 3.97 4.30 3.68 3.46

G-C 3.23 4.1 4.30 3.62 3.45

MRE -0.12 0.60 0.83 0.28 0.10

PBE and revTPSS significantly overestimate the value of rise by 0.6 Å and

0.83Å, MGGA_MS performs much better with 0.28 Å MRE, LSDA and M06L still

work best for the rise value with MARE about 0.1 Å. The larger bond of PBE and

revTPSS is due to the lack of dispersion interaction. By adding the D2 correction to

these functions, the binding energy and position can be greatly improved as shown in

FIG. 4-13.

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FIG. 4-13: The binding curve for PBE, revTPSS, MGGA_MS and D2 correction

functions together with the MP2[99] equilibrium rise value and binding energy.

(Experiment rise at 3.35±0.19 Å)

(ii) Twist and Potential energy surface

In this part, the stacking interactions are studied as a function of the twist angle

between two nucleic bases. The twist between the bases is defined as the higher base’s

right hand rotation around the axe that links the mass center of both higher and lower

bases. (Shown in FIG. 4-14)

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FIG. 4-14: The definition of twist between two bases and four configurations of C-C

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twist from parallel to anti parallel

The initial position of twist or the parallel configuration with twist angle of 0。is

defined as glycosidic bonds are parallel. In the paper of Wu and Yang[102], the

stacking energy was calculated as a function of the twist angel referred to as potential

energy surface (PES) for twist bases, which was chosen as a reliable reference method

to study the interactions between these stacked nucleic bases. Since thymine was not

included in Yang’s work, it is not fully test latterly, thus, not all results of possible

thymine dimers are present here as the lack of reference.

The size of nucleic base is much larger than benzene, as a result, there are

technical difficulties to involve CCSD(T) method to obtain accurate value. Until now,

the most reasonable PES was obtained by employing the MP2/6-31G*(0.25)

method[103], which was confirmed by more accurate studies at anti-parallel

configuration[101,104]. As shown before, MP2 method tends to overestimate the pi-pi

stacking, however, it is showed that this is counteracted by using a medium basis set

with a more diffuse polarization function can lead to only about 1 kcal/mol error with

much less computing cost[88]. In my work, MP2/6-31G*(0.25) results are used to

compare with density functionals.

From the previous result of benzene dimer, the rise for stacked bases, it is

evident that the conventional density functionals considered here cannot well describe

the pi-pi stacking interactions, which should lead to a weak binding ,shallow PES. In

FIG. 4-15, the PES for four stacked systems A-A, C-C, G-G, G-C are presented,

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which are representatives for most possible base pairs.

FIG. 4-15: Stacking energy as a function of twist angle for four stacked bases

computed with various density functionals at the optimized rise and together with the

MP2/6-31G*(0.25) reference data[101].

FIG. 4-15 indicates that PBE and revTPSS cannot correctly reproduce the

stacking energy (either B3LYP, proved by much previous work) while MGGA_MS

predicts quite improved result with respect to MP2. However, at least, the trend and

qualitative features for the energy changing with rotation can be shown by these

functionals with an up shift of the energy. Besides, FIG 4-15 also implies that the

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differences between the maximum and minimum binding energy for these bases

dimers are 6-8 kcal/mol same energy level as hydrogen bond, which suggests that the

pi-pi stacking interaction energy inside the nucleic bases strongly depends on the

twist angle. Configurations with the twist angle close to 0° have the relative small

binding energy due to the overlap orientation leading to largest electrostatic

repulsion, while the configurations with large twist angle have the greater binding

energy but the angles with greatest binding energy differ for various dimers.

It is obvious that the vdW corrections are strongly needed for conventional

functionals to well describe the pi-pi stacking interactions between stacked nucleic

bases. FIG 4-16 shows the corresponding DFT+D2 potential energy surface and it is

evident that they can perform much better with respect to MP2.

FIG 4-16: PES for C-C, G-G calculated with PBE+D2 and revTPSS+D2

Based on these, to well describe the stacking interactions between the nucleic

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bases, the vdW dispersion correction must be included for the classical PBE or

revTPSS functionals. MGGA_MS shows the potential to capture part of this. In the

following section, the interactions between stacked base pairs A:T and G:C are

studied under these functionals.

4.3.3 Stacking interactions for stacked nucleic pairs and

possible reason for the twist of DNA from the view of PES

Previous work in this chapter indicates the stacking interactions strongly

determine the DNA steps rise and twist and a precise description of vdW dispersion

forces is needed for such system. In this part, the DFT+D2 and MGGA_MS

functionals will be used to determine the preferred stacking configurations for four

kinds of W-C DNA nucleic base pairs by studying the stacking energy between them

as a function of rise as well as twist angle. With this, the deeper understanding on the

performance of these functionals for bio-molecules can be gain, the question that what

forces twist DNA can be possibly answered.

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FIG 4-17: Stacking energy at 36° twist calculated with various density

functionals and compared with Sponer et al., in which a standard high-level CCSD(T)

based technique was extrapolated to the complete basis set limit.

The weaker stacking energy leads to large value of rise, which is impossible

inside the DNA, as a result, the energies as a function of rise at the 36° twist angle are

evaluated and compared with the CCSD(T) calculations of Sponer et al[104]. FIG.

4-17 shows the stacking energy for different functionals at their optimized rise and 36°

twist angle and compared with a high-level CCSD(T) calculation. It indicates that for

stacked nucleic base pairs, MGGA_MS can only capture about one third of the

binding energy, which can leads to too long rise vale. On the other hand, the results of

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revTPSS with the D2 correction are consistent with Sponer’s work. These implies that

the effect of long range dispersion forces plays a bigger role in the stacking energy

with larger molecule size and more aromatic rings which can explain why

MGGA_MS performs much worse in stacked pairs than in stacked bases.

Next, we should turn to discuss of the effect of different twist angle on stacking

energy. FIG 4-18 presents the stacking energy of three typical stacked nucleic base

pairs and it indicates that the energy change significantly when twist angle smaller

than 20° and larger than 40°, while there are possible angular stability can be seen

between 25° and 40°. And the final twist angle at 36° should be more influenced by

backbones and solvent in this range.

FIG.4-18: Stacking energy as a function of twist angle for stacked base pairs

AA:TT, GG:CC, AG:CT with revTPSS+D2.

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Based on previous work, it suggests the fact that the stacking interactions

significantly contribute to define the twist phenomenon of DNA: the minima of

stacking interaction energy versus twist angel determine a reasonable range of twist

and the other forces such as backbone-solvent interaction define the final angle.

Furthermore, we can perform energy decomposition analyses based on DFT+D2

results to distinguish the determination of various interactions. As shown in FIG. 4-18,

if we treat the revTPSS energy as the electrostatic (ES) part and D2 correction as the

dispersion interactions part, the sum of them yield the total stacking energy. For

GG:CC pairs, it suggests that although the change in ES is two times larger than the

dispersion part, (17 kcal/mol compared to 8 kcal/mol), neither of them can determine

a stable angle by itself, only by taking into consideration both part of interactions, a

energy minimum can be found.

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FIG. 4-19: Contributions to the revTPSS+D2 total energy as a function of the

twist angle for the stacked GG:CC pairs

4.4 Conclusion

In this chapter, we present the study of hydrogen bond and pi-pi stacking

interaction between nucleic bases systems using a variety of levels of density

functionals, in the absence of contribution from DNA backbones and any solvent. The

hydrogen bonds are the major force connecting the nuclei bases on different

backbones while most density functionals can well describe them, however, adding

vdW correction tend to improve the results. The pi-pi stacking between the stacked

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nucleic base pairs are slightly weaker than hydrogen bonds, but more important in

defining the configuration of pairs by significantly changing the stacking energy with

the different rise and twist angle, as shown in the EPS in FIG. Our results indicates

that this is due to the interplay of mid-long range van der Waals interactions and

electrostatic forces, while the latter one can be well described by DFT, early part,

especially the long-range vdW, is still a hard challenge. Classical functionals, such as

PBE, revTPSS, B3LYP cannot capture these, but the newly developed MGGA_MS

can capture the intermediate part, and leads to much better results but sill misses the

long-range energy with underestimation of the stacking energy. Furthermore, the

DFT+D2 methods are applied to the stacking system with the significant

improvement. Taking into consideration the important role of vdW dispersion forces,

a more accurate method is strongly needed.

In summary, based on the performances for both hydrogen bonds and stacking

interactions, the hybrid functional built with MGGA_MS or revTPSS with an

appropriate long-range vdW correction are good next steps in the development of a

complete density functional for the understanding of more complex molecular

structures.

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Biography

The author received his B.S. in physics at University of Science and

Technology of China in 2007. Then, he was enrolled in the department of physics and

engineering physics to begin his Ph.D. study in 2007. Since then, he has been

studying density functional theory with Professor John P. Perdew.

The following papers are my publications related to this dissertation.

Y. Fang, B. Xiao, J. Tao, J. Sun and J. Perdew, "Ice phases under ambient and high

pressure: Insights from density functional theory", Phys. Rev. B, 87,21401 (2013)

P. Hao, Y. Fang, J. Sun, G. I. Csonka, P. H. T. Philipsen, and J. P. Perdew, “Lattice

constants from semilocal density functionals with zero-point phonon correction”,

Phys. Rev. B, 85, 014111 (2012)

J. Sun, B. Xiao, Y. Fang, R. Haunschild, P. Hao,1 A. Ruzsinszky, G. I. Csonka, G. E.

Scuseria, and J. P. Perdew, "Covalent, Metallic, and Weak Bonds from a Semilocal

Density Functional with the Right Ingredients", Submitted to PRL (2013)