Post Hartree-Fock Methods in Quantum Chemistry

Sourav PalNational Chemical Laboratory

Pune- 411 008

General classification of theoretical chemistry approaches:

Quantum Mechanics (QM) Classical Mechanics (CM)

Molecular Mechanics (MM)

Ab initio methods Semi-empirical methods

Density functional theory (DFT)

THEORETICAL MODEL CHEMISTRY

• Should Include Electron Correlation ( two-electron repulsion effects) in an efficient manner.

• Applicability Should be General

• Results Should Scale Correctly with Number of Electrons N (Size) e.g. energy proportional to N

• Dissociate into Fragments Correctly

• Accuracy; Computationally Tractable ; Ab initio description

Electronic Structure Models• Hartree-Fock Method ( one-particle approximation) sufficient

in many cases; amenable to simple interpretation of molecular orbital theory. The MO theory overweights the ionic parts and thus restricted HF fails to describe dissociation closed shell molecules into open fragments.

• In many cases high level of electron correlation arising from two-electron repulsion needed calling for post-HF rigorous developments

• Configuration interaction, perturbation theory and coupled-cluster methods are the methods of choice

• Among these, coupled-cluster has emerged as a compact method to include correlation and describe size-dependence, dissociation correctly

What is electron correlation and why do we need it?

Recall that the SCF procedure accounts for electron-electron repulsion by optimizing the one-electron MOs in the presence of an average field of the other electrons. The result is that electrons in the same spatial MO (anti-parallel spins) are too close together. Their motion is actually correlated. Correlation of anti-parallel spins missing in Hartree-Fock theory

Eel.cor. = Eexact - EHF (B.O. approx; non-relativistic H)

SD

1 1 2(1) N (1)

1 2 2(2) N (2)

1 N 2(N) N (N)

, i | j ij

Slater Determinant

0 is a single determinantal wavefunction.

Electron Correlation

• Instantaneous repulsion between electrons, missing in mean field or Hartree Fock method

• Correlation between electrons of opposite spins, making them avoid each other

• Virtual orbitals in Hartree-Fock method used for expanding the many-electron wave function in terms of configurations (determinants)

• One way to see why simple MO theory does not work is that at dissociation, more than one determinant is important

• Any post HF method, based on simple RHF method, may not work, in general.

• Post-HF expansion must work on multi-determinant in such cases to correct the problem, in general.

• The state-of-the-art rigorous method is multi-reference coupled-cluster theory, applicable to high accuracy results for any state, away from equilibrium (including dissociation), excited states etc.

• Rigorous method for molecular interaction, properties and reactivity

Size Consistent and Size Extensive

Size-consistent method - the energy of two molecules (or fragments) computed at large separation (100 Å) is equal to the twice energy of the individual molecule (fragment). Only defined if the molecules are non-interacting.

Size-extensive method - the energy scales linearly with the number of particles.

1. Full CI is size consistent and extensive.

2. All forms of truncated CI are not. (Some forms of CI, esp. MR-CI are approximately size consistent and size extensive with a large enough reference space.)

EAB ( R AB) E A + E B

RHF dissociation problem

Consider H2 in a minimal basis composed of one atomic 1s orbital on each atom. Two AOs () leads to two MOs ()…

1 N1(A B ); bonding MO

2 N2 (A B); antibonding MO

H 1s H 1s

H H

H H

The ground state wavefunction is:

0 1(1) 1 (1)

1(2) 1 (2)Slater determinant with two electrons in the bonding MO

0 1 (1)1(2) 1(2)1 (1)

0 1(1)1(2) (1)(2) (1)(2) 0 1(1)1(2)(A (1) B (1))( A (2) B (2))

0 A (1)A (2) B (1)B (2) A (1)B (2) B (1)A (2)

Expand the Slater Determinant

Factor the spatial and spin parts

H does not depend on spin

Four terms in the AO basis

A A

B BIonic terms, two electrons in one Atomic Orbital

A B

B ACovalent terms, two electrons shared between two AOs

H2 Potential Energy Surface

0

E

H H

H. + H.H H

At the dissociation limit, H2 must separate into two neutral atoms.

Bond stretching

At the RHF level, the wavefunction, , is 50% ionic and 50% covalent at all bond lengths.

H2 does not dissociate correctly at the RHF level!! Should be 100% covalent at large internuclear separations.

A A

B B

A B

B A

RHF dissociation problem has several consequences:

• Energies for stretched bonds are too large. Affects transition state structures - Ea are overestimated.

• Equilibrium bond lengths are too short at the RHF level. (Potential well is too steep.) HF method ‘overbinds’ the molecule.

• .

• The wavefunction contains too much ‘ionic’ character; causing dipole moments (and also atomic charges) at the RHF level to be too large.

However, SCF procedures recover ~99% of the total electronic energy around equilibrium.

But, even for small molecules such as H2, the remaining fraction of the energy - the correlation energy - is ~110 kJ/mol, on the order of a chemical bond.

To overcome the RHF dissociation problem,Use two-configurational trial function that is a combination of 0 and 1

Ionic terms Covalent terms

First, write a new wavefunction using the anti-bonding MO.

The form is similar to 0, but describes an excited state:

1 2(1) 2(1)

2(2) 2(2)2(1)2(2) 2 (2)2 (1)

1 2 (1)2(2) (1) (2) (1) (2)

1 2 (1)2(2) (A (1) B (1))( A (2) B (2))

1 A (1) A (2) B (1) B (2) A (1)B (2) B (1)A (2)

2 N2 (A B); antibonding MO

MO basis

AO basis

Configuration interaction

• Linear expansion of wave function in terms of determinants, classified as different ranks of hole-particle excited determinants

• Matrix eigen-value equation• Very accurate determination of a few lowest

eigenvalues using iterative techniques• Problem of proper scaling with size

Trial function - Linear combination of 0 and 1; two electron configurations.

a00 a11 a0(11) a1(22 )

(a0 a1) A A BB (a0 a1) A B BA

Three points:1. As the bond is displaced from equilibrium, the coefficients (a0, a1) vary

until at large separations, a1 = -a0: Ionic terms disappear and the molecule dissociates correctly into two neutral atoms. The above wave function is an example of configuration interaction.

2. The inclusion of anti-bonding character in the wavefunction allows the electrons to be further apart on average. Electronic motion is correlated.

3. The electronic energy will be lower (two variational parameters).

Ionic terms Covalent terms

Configuration Interaction

Since the HF method yields the best single determinant wavefunction and provides about 99% of the total electronic energy, it is commonly used as the reference on which subsequent improvements are based.

As a starting point, consider as a trial function a linear combination of Slater determinants:

a0HF aii1 i Multi-determinant wavefunction

a0 is usually close to 1 (~0.9).

• M basis functions yield M molecular orbitals.• For N electrons, N/2 orbitals are occupied in the RHF wavefunction.• M-N/2 are unoccupied or virtual (anti-bonding) orbitals.

Generate excited Slater determinants by promoting up to N electrons from the N/2 occupied to M-N/2 virtuals:

1

2

3

4

5

6

7

8

9

i

a a a

b b

c

i i

j j

k

HF

ia

ijab

ijkabc

a,b,c… =virtual MOs

i,j,k… = occupied MOs

a,b

i,j

ijab

a

b

c,d

i

j

k,l

ijklabcd

Single Double Triple QuadrupleRef.Excitation level …

Represent the space containing all N-fold excitations by (N).Then the COMPLETE CI wavefunction has the form

Where

CI C0 HF (1) (2) (3) ...(N )

HF Hartree Fock

(1) Ciai

a

a

virt

i

occ

(2) Cijabij

ab

a,b

virt

i, j

occ

(3) Cijkabcijk

abc

a,b,c

virt

i, j ,k

occ

(N ) Cijk...abc...ijk ...

abc...

a,b,c...

virt

i, j ,k...

occ

Linear combination of Slater determinants with single excitationsDoubly excitations

Triples

N-fold excitation

The complete CI expanded in an infinite basis yields the exact solution to the Schrödinger eqn. (Non-relativistic, Born-Oppenheimer approx.), often used as benchmark.

The various coefficients, , may be obtained in a variety of ways. A straightforward method is to use the Variation Principle.

The elements of the vector, , are the coefficients, And the eigenvalue, EK, approximates the energy of the Kth state.

ECI

Cijk...abc... 0

Cijk...abc...

ECI CI | H | CI

CI | CI

HC K EK

C K

Cijk...abc...

C K

Expectation value of He.

Energy is minimized wrt coeff

In a fashion analogous to the HF eqns, the CI Schrodinger equation can be formulated as a matrix eigenvalue problem.

E1 = ECI for the lowest state of a given symmetry and spin.E2 = 1st excited state of the same symmetry and spin, and so on.

Configuration State Functions

Consider a single excitation from the RHF reference.

RHF (1)

Both RHF and (1) have Sz=0, but (1) is not an eigenfunction of S2.

Linear combination of singly excited determinants is an eigenfunction of S2.

Configuration State Function, CSF(Spin Adapted Configuration, SAC)

Singlet CSF

Only CSFs that have the same multiplicity as the HF reference contribute to the correlation energy.

1,2 1(1)2(2) 1 (2)2 (1)

Multi-configuration Self-consistent Field (MCSCF)

1

2

3

4

5

6

7

8

9

HF

H2O MOs

Carry out Full CI and orbital optimization within a small active space. Six-electron in six-orbital MCSCF is shown. Written as [6,6]CASSCF.

Complete Active Space Self-consistent Field (CASSCF)

Why? 1. To have a better description of the ground or

excited state. Some molecules are not well-described by a single Slater determinant, e.g. O3.

2. To describe bond breaking/formation; Transition States.

3. Open-shell system, especially low-spin.4. Low lying energy level(s); mixing with the

ground state produces a better description of the electronic state.

5. …

MCSCF Features:

1. In general, the goal is to provide a better description of the main features of the electronic structure before attempting to recover most of the correlation energy.

2. Some correlation energy (static correlation energy) is recovered. (So called dynamic correlation energy is obtained through CI and other methods through a large N-particle basis.)

3. The choice of active space - occupied and virtual orbitals - is not always obvious. (Chemical intuition and experience help.) Convergence may be poor.

4. CASSCF wavefunctions serve as excellent reference state(s) to recover a larger fraction of the dynamical correlation energy. A CISD calculation from a [n,m]-CASSCF reference is termed Multi-Reference CISD (MR-CISD). With a suitable active space, MRCISD approaches Full CI in accuracy for a given basis even though it is not size-extensive or consistent.

Examples of compounds that require MCSCF for a qualitatively correct description.

C N HC N

H

C NH

Transition State

H

C

H

CH

H

Singlet state of twisted ethene, biradical.

O -

O+

O O

O

O

zwitterionic biradical

Mœller-Plesset Perturbation Theory

In perturbation theory, the solution to one problem is expressed in terms of another one solved previously. The perturbation should be small in some sense relative to the known problem.

ˆ H ˆ H 0 ˆ H '

ˆ H 0 i E i i, i = 0,1,2,...,

ˆ H W

W 0W0 1W1 2W2 ...

00 11 22 ...

Hamiltonian with pert.,

Unperturbed Hamiltonian

As the perturbation is turned on, W (the energy) and change. Use a Taylor series expansion in .

ˆ H 0 ˆ F ii1

N

ˆ h i ˆ J ij ˆ K ij j1

N

i1

N

ˆ H ' gijj1

N

i1

N

gijj1

N

i1

N

W0 sum over MO energies

W1 = 0| | ˆ H ' |0 E(HF)

W2 0| | ˆ H ' | ij

ab ijab | ˆ H ' |0

E0 E ijab

ab

vir

i j

occ

E(MP2) i j |ab i j |ba 2

i j a ba b

vir

i j

occ

Define ˆ H 0 and ˆ H '

Unperturbed H is the sum over Fock operators Moller-Plesset (MP) pert th.

Perturbation is a two-electron operator when H0 is the Fock operator.

With the choice of H0, the first contribution to the correlation energy comes from double excitations.

Explicit formula for 2nd order Moller-Plesset perturbation theory, MP2.

Advantages of MP’n’ Pert. Th.

• MP2 computations on moderate sized systems (~150 basis functions) require the same effort as HF. Scales as M5, but in practice much less.

• Size-extensive (but not variational). Size-extensivity is important; there is no error bound for energy differences. In other words, the error remains relatively constant for different systems.

• Recovers ~80-90% of the correlation energy.

• Can be extended to 4th order: MP4(SDQ) and MP4(SDTQ). MP4(SDTQ) recovers ~95-98% of the correlation energy, but scales as M7.

• Because the computational effort is significanly less than CISD and the size-extensivity, MP2 is a good method for including electron correlation.

Coupled Cluster TheoryPerturbation methods add all types of corrections, e.g., S,D,T,Q,..to a given order (2nd, 3rd, 4th,…).

Coupled cluster (CC) methods include all corrections of a given type to infinite order.

CC e ̂T 0

The CC wavefunction takes on a different form:

Coupled Cluster Wavefunction0 is the HF solution

e ̂T ˆ 1 ˆ T +

12

ˆ T 2 16

ˆ T 3 1k!k0

ˆ T k Exponential operator generates excited Slater determinants

ˆ T ˆ T 1 ˆ T 2 ˆ T 3 ˆ T N Cluster Operator

N is the number of electrons

CC Theory cont.

ˆ T 10 tia

a

vir

i

occ

ia

ˆ T 20 tijab

a b

vir

i j

occ

ijab

The T-operator acting on the HF reference generates all ith excited Slater Determinants, e.g. doubles ij

ab.

tia

tijab

Expansion coefficients are called amplitudes; equivalent to the ai’s in the general multi-determinant wavefunction.

e ̂T ˆ 1 ˆ T 1 ˆ T 2

12

ˆ T 12

ˆ T 3 ˆ T 2ˆ T 1

16

ˆ T 13

ˆ T 4 ˆ T 3ˆ T 1

12

ˆ T 22

12

ˆ T 2ˆ T 1

2 1

24ˆ T 1

4

doubles triples Quadruple excitationssingles

HF ref.

The way that Slater determinants are generated is rather different…

CC Theory cont.

ˆ 1

ˆ T 1

ˆ T 2 12

ˆ T 12

ˆ T 2

ˆ T 12

ˆ T 3 ˆ T 2ˆ T 1

16

ˆ T 13

ˆ T 3

ˆ T 13

ˆ T 4

ˆ T 4 ˆ T 3ˆ T 1

1

2ˆ T 2

2

1

2ˆ T 2

ˆ T 12

1

24ˆ T 1

4

ˆ T 22

ˆ T 2ˆ T 1

HF reference

Singly excited states

Connected doubles

Dis-connected doubles

Connected triples, ‘true’ triples

‘Product’ Triples, disconnected triples

True quadruples - four electrons interacting

Product quadruples - two noninteracting pairs

ˆ T 3ˆ T 1,

ˆ T 2ˆ T 1

2, ˆ T 14 Product quadruples, and so on.

CC Theory cont.

If all cluster operators up to TN are included, the method yields energies that are essentially equivalent to Full CI.

In practice, only the singles and doubles excitation operators are used forming the Coupled Cluster Singles and Doubles model (CCSD).

e ̂T 1 ̂T 2 ˆ 1 ˆ T 1 ˆ T 2

12

ˆ T 12

ˆ T 2ˆ T 1

16

ˆ T 13

12

ˆ T 22

12

ˆ T 2ˆ T 1

2 1

24ˆ T 1

4

The result is that triple and quadruple excitations also enter into the energy expression (not shown) via products of single and double amplitudes.

It has been shown that the connected triples term, T3, can be important. It can be included perturbatively at a modest cost to yield the CCSD(T) model. With the inclusion of connected triples, the CCSD(T) model yields energies close to the Full CI in the given basis, a very accurate wavefunction.

Comparison of Models

Accuracy with a medium sized basis set (single determinant reference):

HF << MP2 < CISD < MP4(SDQ) ~CCSD < MP4(SDTQ) < CCSD(T)

CI-SD CI-SDTQ MP2 MP4(SDTQ) CCSD CCSD(T)Scale with M M6 M10 M5 M7 M6 M7

Size-extensive/consistent No ~Yes Y Y Y YVariational Y Y No No No No

Generally applicable Y No Y Y Y YRequires ŌgoodÕ zero-order Y ~No Y Y ~No NoExtension to Multi-reference Yes Yes Not yet

common

In cases where there is (a) strong multi-reference character and (b) for excited states, MR-CI methods may be the best option.

Size-dependence

• Dimer of non-interacting H2 molecules

• D-CI wave function contains doubly excited determinants on each H2 molecule , but not the quadruply excited determinant

• The product of monomer D CI, however, contains the quadruply excited

• Φab x Φcd

• However, exponential wave function of dimer even at doubles level contains quadruply excited determinant

MOLECULAR PROPERTIES

• Defined as derivatives of molecular energy with respect to perturbation parameters.

EXAMPLES: DIPOLE MOMENT, POLARIZIBILITY, NUCLEAR GRADIENTS AND HESSIANS ETC.,

• Finite-field (numerical) method.

– Numerical evaluation of energy derivatives by computing energy at least two different fields.

– Evaluate (no extra effort other than calculating energy)

– Very inaccurate as it involves taking difference of two large. numbers.

• ANALYTIC METHOD

– Closed analytic form for energy derivatives.

– Requires extra effort to solve for energy

– Response equations.

– easier evaluation of energy and property surfaces.

• Properties for variational (stationary) theories

– S.Pal, TCA 66, 151(1984); PRA 34, 2682(1986); PRA 33, 2240(1986); TCA 68, 379(1985); Vaval, Ghose and Pal, JCP 101, 4914 (1994); Vaval & Pal, PRA 54, 250(1996).

Coupled-Cluster Response• H()=H+ O

• Equations for different order response:

• E (1) = < | exp(-T) {O +[H,T (1) ]} exp (T)| >• 0 = < * | exp(-T) {O+[H,T (1) ]} exp (T)| >

• Linear Equation to be solved for T (1) amplitudes

• Due to multi-commutator expansion, extensivity of properties retained

• Properties with coupled cluster method

– Non-variational theory

– No Hellmann-Feynman theorem

– Energy first derivative depends on first derivatives of cluster amplitudes which have to be explicitly computed.

E(1) = YT T(1) + …. y T,A (Perturbation independent)

AT(1) = B B Perturbation independent

Z-VECTOR METHOD FOR SRCC

–Recasting energy derivative expression to eliminate perturbation dependent cluster derivative in favour of perturbation independent z-vector

E(1) = yTA-1 B = ZT B ( ZT set of de-excitation amplitudes)+..

Z-vector needs to be solved only once.

ZTA = yT (Perturbation independent)

Dalgarno and Stewart, Handy and Schaefer

Z-Vector can be introduced by stationary approach using Lagrange multipliers

£ = E - ZiT < i

* | exp(-T) H exp (T)| >

£ / T = 0 provides Z amplitudes same as non-var

Case of near-degeneracy

• At away from equilibrium, more than one configuration is important (non-dynamic correlation)

• Perturbative or cluster expansion around any one determinant causes convergence problem

• Coupled-cluster equations based on single reference suffer from convergence (intruder)

• A correct way to solve the problem is to start from the a space of important determinants and the exponential wave-operator to generate dynamic correlation

• Several different versions of multi-reference CC theory• Hilbert and Fock space, State-selective type

Multi-reference coupled-cluster• For near-degenerate systems, methods with starting space

consisting of a linear combination of multi-determinants and an exponential operator acting on this space constitute a class of multi-reference coupled-cluster theories

• Very powerful method with proper combination of dynamic and non-dynamic electron correlation

• The nature of T operator can vary in this case, depending on the nature of the MRCC method.

• Multiple roots obtained by diagonalisation of an effective Hamiltonian over the starting model space

• Contribution to the Fock space MRCC, ideal for ionization/ excitation energies etc.

• (Mukherjee D and Pal S, Adv Quant Chem 20, 291 (1989); Pal et al, J. Chem. Phys. 88, 4357 (1988); Vaval and Pal, J. Chem. Phys. 111, 4051 (1999)

MULTI-REFERENCE THEORIES : EFFECTIVE HAMILTONIAN

APPROACH

• Define quasi-degenerate model space p

• P° = l> < | |(0) = Ci |

• Transform Hamiltonian by ‘’ to obtain an effective Hamiltonian such that it has same eigen values as the real Hamiltonian.

P° Heff P° = P°H P° (Heff)ij Cj = ECi

• Obtain energies of all interacting states in model space by diagonalizing effective Hamiltonian over small dimensional model space p

MULTI-REFERENCE THEORIES : EFFECTIVE HAMILTONIAN

APPROACH• Bloch equation

H = Heff

• Coupled cluster anastz for wave operator = exp(T)

• P[H - Heff ] P = 0 Q[H - Heff ] P = 0Heff C = C E

• Multiple states at a time at a particular geometry

Variants of Multi-reference CC• Effective Hamiltonian theory: Effective Hamiltonian over the

model space of principal determinants constructed and energies obtained as eigen values of the effective Hamiltonian

• Valence-universal or Fock space: Suitable for difference energies ( Mukherjee, Kutzelnigg, Lindgren, Kaldor and others)

Common vacuum concept; Wave-operator consists of hole-particle creation, but also destruction of active holes and particles contained in the model space

• State-universal or Hilbert space: Suitable for the potential energy surface. Each determinant acts as a vacuum (Jeziorski and Monkhorst; Jeziorski and Paldus ; Balkova and Bartlett)

Multi- reference coupled cluster thus is more general and powerful electronic structure theory

To make the theory applicable to energy derivatives like properties or gradients, Hessians etc., it is important to develop linear response to the MRCC theoryEqn for Cluster amplitude derivative and Heff

= / { P[H - Heff ] P} = 0 = / { Q[H - Heff ] P} = 0

Eqn. For energy derivative and model space coefficient derivativeHeff

(1) C + Heff C(1) = C (1) E + C E(1)

S. Pal, Phys. Rev A 39, 39, (1989); S. Pal, Int. J. Quantum Chem, 41, 443 (1992)

Fock Space Multi-reference Coupled-Cluster Approach

• ( Mukherjee and Pal, Adv. Quant. Chem. 20, 291 ,1989)

• N-electron RHF chosen as a vacuum, with respect to which holes and particles are defined.

• Subdivision of holes and particles into active and inactive space, depending on model space

• General model space with m-particles and n-holes

(0) (m,n)

= iC i i

(m,n)

Fock space MRCC• P(k,1)[H - Heff ] P(k,1) = 0 k = 0, m; 1=0, n

Q(k,1)[H - Heff ] P(k,1) = 0 k = 0, m; 1=0, n

Heff is the effective Hamiltonian defined over the model space determinants. The eigen values of it gives the exact energies of interest.

• For low-lying excited states one hole one particle model space is suitable.

Fock Space MRCC• For the general one active hole and one active particle problem

the model space is written as

|µ(0)(k,1)> = iCi |i(k,1)> k=0,1; l=0,1

The wave operator will be {exp ( S(k,l)) }• S (k,1) = Ť(0,0) + Ť(0,1) + Ť(1,0) + Ť(1,1) • For the (1,1) problemFor the (1,1) problem,, the model space is an incomplete model the model space is an incomplete model

space (IMS). Though generally for IMS, to have linked cluster space (IMS). Though generally for IMS, to have linked cluster theorem, intermediate normalization has to be abandoned, for theorem, intermediate normalization has to be abandoned, for (1,1) model space, the equations can be derived assuming (1,1) model space, the equations can be derived assuming intermediate normalization.intermediate normalization.

PP(1,1)(1,1) P P(1,1)(1,1) P P(1,1)(1,1) + P + P(1,1)(1,1) T T11 (1,1)(1,1) T T11 (0,0)(0,0) P(1,1) +…… P(1,1) +……

In addition, for (1,1) model space T1(1,1) operator is

in the wave operator, but it does not contribute to the energy and thus can be neglected in the energy derivative or linear response problem

Computationally full singles and doubles approximation has been used. For excitation energies closed part of the connected (H exp(T(0,0)) is dropped, to facilitate direct evaluation.

Finally to get the singlet and triplet excited states one diagonalizes the spin integrated effective Hamiltonian matrices HS

EE and HTEE

Z- Vector method for MRCC theory• In a compact form the response equation may be written as,

• A T (1) = B

• A : Perturbation -independent matrix

• B : Perturbation-dependent column vector

Two options: Eliminate T(1) in Heff (1)

• Eliminate perturbation-dependent T(1) in energy expression

Elimination in Heff (1) ensures that all roots can be obtained

without using perturbation-dependent vectors, but this requires a larger no of Z vectors ( square of model space)

D. Ajitha, N. Vaval and S. Pal, J Chem Phys 110, 8236 (1999); J. Chem. Phys 114, 3380 (2001);

Z-vector solved from a perturbation independent linear equation

Use elimination in energy expression for a single root,

E I (1) = C' i [Heff (1) ] Ci

Simplified expression

E I (1) = Y (I) * T (1) + X(I) * F(1) + Q(I)* V(1)

Define Z-vector Z(I)through Matrix equation Y (I) = Z (I) A

E I (1) expressed in terms of z-vector

E I (1) = Z ( I) * B + X (I) F (1) + Q (I) * V (1)

Z - vector although perturbation independent, still depends on state of interestNo single Z- vector for all roots at the same time

K R Shamasundar and S. Pal, J. Chem. Phys. 114, 1981 (2001); Int. J. Mol. Sci. 3, 710 (2002)

Analytic linear response

• Analytic linear response for FSMRCC H(g) = H + gH(1)

Θ = {T, Heff, E, C, Č} perturbation dependent Θ(g) = Θ(0) + g Θ(1) + ½!g2 Θ (2) + ….. Θ(n) :: nth order responseHierarchical equations for response quantities Θ(n) can be derived.

Specific expressions derived for [0,1], [1,0] and [1,1] sectors and first order response of energy calculated.

C,Č obey (2n+1) – rule, but the same advantage not enjoyed by the cluster amplitudes.

Z-vector response approach to FS MRCC• Early attempt:- Use of a de-excitation vector Z of same

size as total T amplitudes T[0,0] and T[0,1]. Factorization of response equation possible only for T[0,1](1) and is only for the highest valence case.

For HSMRCC M2 linearly independent de-excitation amplitudes can eliminate totally all elements of T(1). Elimination of T(1) can be carried out separately for each element E(1)

Dependent on C, Č State specific

• Shamasundar and Pal ; JCP 114, 1981 (2001)

Constrained Variation

• In SRCC, the Z-vector can be introduced by making energy stationary with non-variational CC equations as constrains ( Jorgensen et al)

• Along the lines of SRCC, same can be effected by using variation of ČHeff C with constraints on the equations for T( Lagrange multiplier).

• The approach used for Hilbert space MRCCand Fock space MRCC

• Shamasundar and Pal, Int. J Mol. Sci. 4, (2003)

• Shamasundar, Asokan and Pal J. Chem. Phys. 120, 6381 (2004)

Structure of FSMRCC response equations

[ ][ ] [ ] [ ] [ ] [ ] [ ]

0

[ ] [ ]

( ) , ,

1

( )

( )

nnn n i i i i

A A eff effi

n nA A A

C C H T H

E C C

J MQ

[ ][ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

,

[ ][ ] [ ] [ ] [ ] [ ]

,

, , | | | |

| | | |

( )i

i i i i i i i i ieffeff l l

l

ii i i i i

eff

T H H H

H H

M

Structure of FSMRCC response equations

• The stationary equations are obtained by making the Lagrange functional stationary with respect to the T amplitudes, amplitudes and effective Hamiltonian

elements.

[ ]

[ ]0

kn

ik i T

M

[ ]

[ ][ ] [ ] 0i

knn n ni A A

k i eff

C CH

M

Structure of FSMRCC response equations

[ ]

[ ][ ] [ ] 0i

knn n ni A A

k i eff

C CH

M

Structure of FSMRCC response equations• M (k) depends on lower-valence T’s. Hence in the

stationary equation with respect to T[i] summation index is from i to n , where n is the highest valence sector.

• To solve Lagrange multipliers for a specific sector, all higher valence ’s are necessary.

• SEC decoupling, reverse to the T equations, is present in the equations

• In the [i] determining equation, all higher valence are present and are in the inhomogeneous part of the equation

• For the equation determining highest valence , the inhomogeneous part contains model space coefficients and this makes the theory state-specific

Structure of FSMRCC response equations

• For the highest valence sector, the non-zero inhomogeneous parts are present only for the closed parts of (n)

• Open and closed parts of are coupled in each FS sector• Closed part of Lagrange multipliers takes care of

incompleteness of model space• For incomplete model space, the closed parts appear

explicitly because effective Hamiltonian can not be defined explicitly in terms of the cluster amplitudes

• Similarity transformation approach can eliminate the closed parts, but this is not available in general

Structure of FSMRCC response equations

• Special case of incomplete model space/ complete model space (CMS) results in simplifications. For CMS, intermediate normalization makes the definition of the effective Hamiltonian explicit, thus allowing definition of closed parts of in terms of open amplitudes of , T and model space coefficients

• P H P = P Heff P = P Heff P• This the solution of amplitudes involves only the open

parts of them• Similar simplifications appear for quasi-complete model

space e.g. (1,1) Fock space • For (1,1) model space, the one-body part of (1,1) T and

amplitudes are not specifically required for energy derivatives, since connected diagrams are not possible using these operators.

Rigorous spectra and properties

• Accurate calculation of difference energies using multi-reference coupled-cluster method (accuracy within 0.1mev)

• Developed analytic approach based on variational coupled cluster method for molecular properties of closed and open shell molecules, excited states.

S. Pal, M.Rittby , R.J.Bartlett, D.Sinha and D.Mukherjee J.Chem.Phys.,88,4357 (1988); N. Vaval, K.B.Ghose, S. Pal and D.Mukherjee, Chem.Phys.Lett.,209, 292 (1993); N. Vaval and S. Pal, J. Chem. Phys 111, 4051(1999); S.Pal, Theor. Chim. Acta., 68, 379 (1985); Phys.Rev.A, 33, 2240 (1986); Phys. Rev. A. 34, 2682 (1986); Phys.Rev.A,39,39,(1989); N. Vaval, K.B.Ghose and S. Pal, J. Chem. Phys, 101, 4914, (1994); N. Vaval & S.Pal, Phys.Rev.A 54, 250 (1996); D. Ajitha, N.Vaval and S. Pal, J. Chem. Phys. 110, 2316 (1999); P. Manohar and S.Pal, Chem.Phys.Lett. (2007) (In Press)

TABLE I. Adiabatic excitation energies and dipole moments of H2O using the FSMRCC response approach

State Excitation energy

(eV)

Expt. Excitation energy (eV)

Total energy at the FSMRCC in a.u.

MRCC FF (a.u.)

MRCC Anal (a.u.)

CASSCPa (a.u.)

1B1(1b13sa1)

7.287 7.49b -76.0221

(0.005 a.u.)

-76.0247

(0.00 a.u.)

-76.0284

(-0.005 a.u.)

-0.636 -0.603 -0.712

3B1(1b13sa1)

6.878 7.0,c7.2d -76.0374

(0.005 a.u.)

-76.0397

(0.00 a.u.)

-76.0426

(-0.005 a.u.)

-0.520 -0.599 -0.478

Table II: Dipole moment values of the HCOO radical using analytic Fock space multi-reference

coupled cluster response approach

Manohar, Vaval and Pal , Theo. Chem., 768, 91 (2006)

Table III: Dipole moment of OH radical (in au) using FSMRCC method

Manohar, Vaval and Pal , Theo. Chem., 768, 91 (2006)

Table IV: Dipole moment of Nitrogen oxides (in au) using FSMRCC method

Higher-order energy Derivatives

• £ (g, ) = £ (0) + g £ (1) + 1/ 2 gg £ (2) +…..

• £ (n) is a functional of quantities upto (m) (m=0,1,2 ..n).

• Response of the quantities are obtained by making (m) stationary with respect to (0) i.e.

£ (n) / =0

For first-order response, £ (1) / =0, yielding response quantities.

Use (2n+1) type rule to explicitly compute higher energy derivatives upto third-order

Structure of first-order response wave-function

Sector wise solution of the (1) amplitudes Similar structure of Fock space equations for first

derivatives of T and hold good. (SEC decoupling starting from lowest sector for T(1) amplitudes and reverse SEC decoupling for (1) amplitudes

Use (2n+1) type rule to explicitly compute higher energy derivatives upto third-order

Expressions derived upto (1,1) sector.

Table V Polarizabilities of OH, HCOO and OOH radicals

Manohar and Pal, communicated

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For

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