1

19. Population Analysis

Population analysis is the study of charge distribution within molecules. The intention is to

accurately model partial charge magnitude and location within a molecule. This can be thought of

as a rigorous version of assigning partial charges on the atoms like chemists often do in Lewis dot

structures. Partial atomic charges are not observable characteristics of molecules, and therefore the

entire idea of modeling electron population is not unique. In order to assign charges to atoms, one

must define the spatial region of those atoms, then add up all the charge in that region (integrate

charge density over the volume). This may sound trivial, but where does one atom’s electron cloud

end and the next begin? Which nucleus does a specific region of the electron cloud belong to? The

process is arbitrary, but the results are useful.

19.1. Mulliken population analysis

Mulliken population analysis is by default always performed in Gaussian. It is based on the linear

combination of atomic orbitals and therefore the wave function of the molecule. The electrons are

partitioned to the atoms based on the nature of the atomic orbitals’ contribution to the molecular

wave function. Generally, the total number of electrons in the molecule N can be expressed as:

𝑁 = ∑ ∫ 𝜓𝑗(𝑟𝑗)𝜓𝑗(𝑟𝑗)𝑑𝑟𝑗𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠𝑗

= ∑ ∑ ∫ 𝑐𝑗𝑟𝜑𝑟(𝑟𝑗)𝑐𝑗𝑠𝜑𝑠(𝑟𝑠)𝑑𝑟𝑗𝑟,𝑠𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠𝑗 (152)

= ∑ (∑ 𝑐𝑗𝑟2 + ∑ 𝑐𝑗𝑟𝑐𝑗𝑠𝑆𝑟𝑠

𝑟≠𝑠

𝑟

)

𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠

𝑗

Where r and s index the AO basis functions cjr are coefficients of the basis function r in the MO

j, and S is the overlap matrix defined before. This shows that the total number of electrons can be

divided into two sums: the first one including only squares of single AO basis function (r), and the

other one products of two different AO functions (r and s). Clearly the first term can be thought of

as electrons belonging to the particular atom. It is the second term that causes problems – there is

no single best way how to divide the shared electrons between the two atoms.

Mulliken suggested to split the shared density 50:50. Then the electrons associated with the atom

k are given by:

𝑁𝑘 = ∑ (∑ 𝑐𝑗𝑟2 + ∑ 𝑐𝑗𝑟𝑐𝑗𝑠𝑆𝑟𝑠𝑟,𝑠∈𝑘,𝑟≠𝑠 + ∑ 𝑐𝑗𝑟𝑐𝑗𝑠𝑆𝑟𝑠𝑟∈𝑘,𝑠∉𝑘𝑟∈𝑘 )𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠

𝑗 (153)

where the first two terms come from the basis functions on the kth atom and the last term is the

part shared with all other atoms. The partial charge on the atom k is then:

𝑞𝑘 = 𝑍𝑘 − 𝑁𝑘 (154)

where Zk is its atomic number.

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Advantages: This theory is available for use in nearly every software program for molecular

modeling, and is computationally cheap. For minimal and small split valence basis sets, this

method quickly gives chemically intuitive charge sign on atoms and usually reasonable charge

magnitudes. It works well for comparing changes in partial charge assignment between two

different geometries when the same size basis set is used.

Disadvantages: The partial charges assigned to atoms using Mulliken population analysis vary

significantly for the same system when different size basis sets are used, so computations using

different basis sets cannot be compared. If non-orthonormal basis sets are used, individual basis

functions can have occupation numbers greater than 1 (or 2 in restricted theories), which is

physically meaningless. This instability of charge with increasing basis set size is a major

disadvantage to the theory. There is no way to account for differences in electronegativities of

atoms within the molecule; the method always equally distributes shared electrons between two

atoms.

19.2. Lwdin population analysis:

The Lowdin population analysis method sought to improve upon the Mulliken method, mostly to

correct the instability of predicted charges with increasing basis set size. This is achieved by

transforming the atomic orbital basis functions into an orthonormal set of basis functions prior to

the population analysis. Orthonormal means that they do not overlap (the overlap - S - is zero).

Therefore, the transformations eliminates the overlap term and with it the need for deciding what

to do with it. After proper transformation the formula for Nk from Lӧwdin analysis would

correspond to that for the Mulliken one above, but only with the first term: the other two would be

zero due to orthonormalizaiton.

Advantages: Lӧwdin population analysis is more stable than Mulliken with changes in the basis

set, although with very large basis sets it may also have problems.

Disadvantages: This method uses a symmetric orthogonalization scheme of the atomic orbitals,

and still does not account for electronegativity of different atoms. It is more computationally

expensive than Mulliken analysis. Unfortunately it appears there is no way to use Lӧwdin’s method

with our software.

19.3. Natural Bond Orbitals (NBO):

To use NBO for electron population analysis in Gaussian, include Pop=NPA in the input file. NPA

stands for Natural Population Analysis and is based on the Natural Bond Orbital (NBO) scheme

(the complete NBO analysis can be done using Pop=NBO)

Natural bond analysis classifies and localizes orbitals into three distinct groups: non-bonding

natural atomic orbitals (NAOs), orbitals involved in bonding and antibonding (NBOs), and

Rydberg type orbitals. The Rydberg type orbitals and NAOs are made up of basis sets of single

atoms and the NBOs are a combination of basis set atomic orbitals of two atoms. This is similar to

our notion of core electrons, lone pairs of electrons, and valence electrons, and works under the

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assumption that only the bonding orbitals should be made by combinations of two atoms’ basis

sets. Based on this model of electron partitioning, Natural Population Analysis then treats the

NBOs as the Mulliken method treats all the orbitals.

Advantages: This method differentiates between the orbitals that will overlap to form a bond and

those that are too near the core of an atom to be involved in bonding. This results in convergence

of atomic partial charge to a stable value as the basis set size is increased.

Disadvantages: More computationally expensive than both Mulliken and Lӧwdin methodologies.

NPA also tends to predict larger charges than several other population analysis methods, so like

Mulliken charges NPA is best used for comparing differences rather than determining absolute

atomic charges.

19.4. Atoms in Molecules (AIM):

The population analysis using Atoms in Molecules theory is requested by keyword AIM in a

Gaussian input file. Note that you don’t find this in the manual (or Gaussian website).

The notion of an atom in the molecule, and therefore the location of the partial charges

seems trivial, but how do we define the atom within a molecule? It is not a point, nor is it a clearly

defined sphere. The electron cloud is diffuse, and the exact solution to the Schrödinger equation

requires us to consider all space, so how to define the spatial existence of an atom is not so trivial.

Atoms in Molecules theory creates the spatial partition of atoms with zero-flux surfaces of the

electron density. Electron density is measureable, though AIM bases its calculations on the

calculated electron density. The zero flux surfaces define the spatial region over which the electron

density is integrated to determine total charge in an atom’s “space”. The existence of critical points

(where the derivative of the flux surface is zero) defines the existence of a bond between two nuclei

in AIM. The zero-flux surface can be defined mathematically as the union of all points for which

∇𝜌 ∙ 𝒏 = 0 where 𝜌 is the electron density function (from the wave function) and n is the unit

vector normal to the surface. Once the zero-flux surfaces (and therefore the spatial region Ω𝑘 for

an atom) are calculated, the partial charge on the atom is computed as:

𝑞𝑘 = 𝑍𝑘 − ∫ 𝜌(𝑟)𝑑𝑟Ω𝑘

(155)

The image below is from Lewars: Computational chemistry: introduction to the theory and

applications of molecular and quantum mechanics (Springer, 2011). It shows the trajectories of

the gradient vector field of electron density (the arrows), A and B are two nuclei, the dark line

labeled S is a slice of the zero-flux surface that defines the spatial region of each atom, and the

point C is the bond critical point.

4

Advantages: The partitioning of an atom’s volume makes sense, and this method does take

electronegativity into account. This method is independent of basis set as well as the calculation

method (level of theory).

Disadvantages: This method produces partial charges that are often seemingly odd. For example,

a saturated hydrocarbon is assigned weakly positive carbons and weakly negative hydrogens, not

what every other method for computing partial charges predicts. The odd behavior is rooted in the

non-uniform distribution of charge within the AIM definition of an atoms space. This method is

also relatively computationally expensive. As a result, this method is not often used.

The above image (also from Lewars) provides a good visualization for the meaning of a bond

critical point. It resembles the transition state structure of a PES, but the “surface” is the inverse

of electron density.

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19.5. Population analyses base on the Molecular Electrostatic Potential

The molecular electrostatic potential (MEP) is observable and is used for both Merz-Kollman and

ChelpG (CHarges from ELectrostatic Potentials using a Grid based method) population analysis

methods. These methods are requested through Pop=MK and Pop=ChelpG, respectively. Though

the MEP is observable (can be measured) in the population analysis it is computed from the wave

function of the system. The most notable disadvantage to both MK and ChelpG is that they do not

work well for large systems, especially those with “inner” atoms because the MEP is a surface

characteristic.

The ChelpG and MK population analysis assign atomic partial charges such that the MEP

is replicated. The quality that sets the two methods apart is that with ChelpG the MEP is calculated

along points that are spaced 3.0picometers apart and lie along a cube that encompasses the

molecule. Any points lying on this cube but within the van der Waals surface are discarded. The

dimensions of the cube are such that the closest the cube is to the molecule is 28.0 picometers.

Once the cube is formed, and the MEP has been calculated at points on the cube, the atomic charges

are assigned such that the MEP is reproduced.

The MK population analysis method calculates the MEP along 4 layers encompassing the

molecule, each layer is scaling factor larger than the van der Waals surface (layer one is at 1.4 the

van der Waals radii, layer two at 1.6, layer three at 1.8, and layer four at 2.0). The molecular

electrostatic potential is computed at many points along the layers from the wave function, and the

electron distribution is then made to replicate the molecular electrostatic potential. The number of

layers can be increased by calling for it in the input file.

19.6. Comparing methods:

The following table compares the calculated charge on the hydrogen atom of HF using

several sized basis sets and three different population analysis methods. The basis set size

dependence of the Mulliken method is apparent, while for the electrostatic (MK and ChelpG) and

NBO methods there is significantly less variation in predicted charges as basis set size is changed.

6

20. NMR Spectroscopy and Magnetic Properties

Nuclear Magnetic Resonance (NMR) Spectroscopy is a technique used largely by organic,

inorganic, and biological chemists to determine a variety of physical and chemical properties of

atoms and the molecules they are part of. NMR instruments use a powerful magnet to create a

strong magnetic field. Nuclei that are spin active, such as protons or carbon-13 nuclei, absorb

electromagnetic radiation at a frequency specific to that isotope.

One dimensional NMR spectra display a number of signals (peaks) equivalent to the

number of inequivalent groups of one kind of nucleus. For example, methane has four hydrogens

(protons), but they are all equivalent. This means that if a proton is exchanged for another group,

it is impossible to tell which of the four protons was removed. The proton spectrum for methane

is thus one single peak.

20.1. NMR Shielding

Shielding refers to have guarded a nucleus is against the magnetic field created by the NMR

magnet. As electron density surrounding a nucleus increases, that nucleus becomes more shielded

against the magnetic field. Conversely, nuclei are deshielded as electron density decreases. This

results in a shift in the placement of a peak on the x-axis of an NMR spectrum. The more to the

right a peak appears, the more shielded that nucleus is. The more to the left a peak appears, the

more deshielded that nucleus is. Do note that shifts can be negative, but nearly all carbon-13 and

proton peaks will appear at a positive ppm.

The x-axis of NMR spectra is in parts per million, ppm. This is actually the ratio of the frequency

of the spin active nucleus in the magnet to the frequency of the magnet itself multiplied by one

million. The University of Wyoming has a 400 MHz NMR and a 600 MHz NMR. If the resonant

frequency of a proton is 400 Hz on the 400 MHz NMR, then (400 Hz/400 MHz)*106 = 1 ppm.

The same proton in the same molecule on the 600 MHz NMR would still appear extremely close

to 1 ppm, but its resonant frequency would now be approximately 600 Hz. Using ppm instead of

simply hertz allows chemists to use different magnet strengths, but still compare each other's data

7

because equivalent nuclei always give the same chemical shift in ppm regardless of the magnet

strength.

Adding electronegative groups, such as halogens, singly bonded oxygens, or doubly bonded

oxygens, to a carbon atom removes electron density from that carbon and from any protons still

bonded to the carbon. By removing electron density from these nuclei, the nuclei are less shielded

from the magnetic field and will be deshielded. They will appear at a higher ppm than a similar

hydrocarbon.

Chemical shifts are always relative to a standard molecule, most of that

of tetramethyl silane, TMS. Outputs of calculations will give absolute

chemical shifts, but in order to compare them to experimental values,

calculate the absolute chemical shift for TMS and subtract the absolute

chemical shift of your specified molecule from the absolute chemical

shift of TMS.

20.2. Predicting NMR Properties

1. Chemical Shifts. Simply adding "NMR" in the route section will allow shielding values to be

calculated.

#T RHF/6-31G(d) NMR Test

You should optimize the geometry first as always. NMR calculations benefit greatly from accurate

geometries and larger basis sets. The following is an example calculation for the chemical shift of

the carbon nucleus in methane:

%Chk=NMRmethane

#T B3LYP/6-31G(d) Opt Test

Opt

Tetramethylsilane (TMS)

8

0 1

C 0 -0.0004000000 0.0006000000 0.0002000000

H 0 -0.0754000000 1.0126000000 0.3412000000

H 0 1.0316000000 -0.2674000000 -0.0988000000

H 0 -0.4694000000 -0.6514000000 0.7062000000

H 0 -0.4864000000 -0.0944000000 -0.9488000000

--Link1--

%Chk=NMRmethane

#T RHF/6-31G(d) NMR Geom=Check Guess=Read Test

NMR

0 1

The output will include the following:

GIAO Magnetic shielding tensor (ppm):

1 C Isotropic = 199.0522

The chemical shift of experimental methane does not remotely match this chemical shift, however,

because this is an absolute chemical shift (or, more precisely, the absolute value of the magnetic

shielding). We need to calculate the absolute chemical shift of TMS next. Of course, it has to be

calculated at the same level of theory – by now this should be obvious why.

The TMS output will appear as follows:

GIAO Magnetic shielding tensor (ppm):

1 C Isotropic = 195.1196

By subtracting the value of the standard (TMS) from the one for your sample (methane), the

desired chemical shift is obtained;

195.1196 - 199.0522 = -3.9 ppm

The experimental value is -7.0 ppm. Note that a negative sign indicates that the specified molecule

is more shielded than the reference molecule while a positive sign indicates that the specified

molecule is less shielded than the reference molecule. All experimental chemical shifts are

relative.

2. Spin-Spin Coupling. Nuclei themselves possess a small magnetic field and can therefore

influence the frequency of nearby nuclei. This pair of nuclei are said to be coupled. Coupling

constants are expressed in Hz and typically range from a few hertz up to 20 Hz. The most common

type of coupling is scalar coupling which occurs through chemical bonds. Two nuclei are less

likely to be coupled as the number of bonds between them increases, with coupling between nuclei

more than three bonds apart being fairly rare.

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Splitting patterns result from coupling in proton NMR spectra. When a proton is note coupled to

any other proton, it appears as a singlet: one tall peak. When a proton is coupled to one and only

one other proton, the peak appears as a doublet: two identical, or nearly identical, peaks in terms

of height and peak area. When a proton is coupled to two equivalent protons, its peak appears as

a triplet: three peaks with a height ratio of 1:2:1. Simple splitting patterns follow the n+1 rule,

where n is the number of equivalent protons that a proton is split by. The heights of the peaks that

result follow patterns present in Pascal's Triangle, ie doublets are 1:1, triplets 1:2:1, quartets

1:3:3:1, etc.

If a proton is split by two or more groups of equivalent protons, for example protons on carbon 2

of n-pentane, the splitting pattern becomes much more complicated.

The protons in green are not coupled to any other proton, as the closest non-equivalent proton is 5

bonds away. They appear as a single peak, a singlet, as a result. The red protons are coupled to

the three blue protons, so the red protons appear as a quartet with a 1:3:3:1 relative height ratio.

The blue protons are split only by the 2 red protons, so the blue protons appear as a triplet.

The coupling constant(s) for a peak, if it has any, is the distance in Hz that a peak was split into.

To calculate coupling constants, add:

NMR=SpinSpin

to the route section.

If SpinSpin is included in the previous methane calculations, an additional output is included as

follows:

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Total nuclear spin-spin coupling J (Hz):

1 2 3 4 5

1 0.000000D+00

2 0.149388D+03 0.000000D+00

3 0.149289D+03 -0.307537D+02 0.000000D+00

4 0.149336D+03 -0.307217D+02 -0.307820D+02 0.000000D+00

5 0.149330D+03 -0.307364D+02 -0.307719D+02 -0.307599D+02 0.000000D+00

20.3. Technical Issues with Calculating NMR Properties

One of the problems with calculating magnetic properties, including NMR, is the so-called

origin-gauge dependence. That means that what we calculate is generally dependent on where we

pick the origin of our coordinate system. Obviously, the real properties cannot depend on that. The

reason why the calculated properties change with the selection of the gauge origin is the

approximate, not exact, wave function and the finite basis set. It can be shown that in the limit of

an infinite bases set the properties become gauge invariant. Unfortunately, working with an infinite

basis is hardly an option.

To reduce artifacts associated with the gauge origin, two different approaches have seen

extensive use in the literature. The older method employs gauge-including atomic orbitals

(GIAOs) as a basis set. By a clever incorporation of the gauge origin into the basis functions

themselves, all matrix elements involving the basis functions can be arranged to be independent

of it. An alternative is the ‘individual gauge for localized orbitals’ (IGLO) method, where different

gauge origins are used for each localized MO in order to minimize error introduced by having the

gauge origin far from any particular MO. Of the two methods, modern implementations of GIAO

are probably somewhat more robust, but it is possible to obtain good results with either.

One should also be aware of issues with using the Effective Core Potentials (ECPs). If

the core electrons of the heavy atom are represented by an ECP, then it is not in general possible

to predict the chemical shift for that nucleus, since the remaining basis functions will have incorrect

behavior at the nuclear position (note that it is mostly the ‘tails’ of the valence orbitals at the

nucleus that influence the chemical shift, not the core orbitals themselves, since they are filled

shells). However, ECPs may be an efficient choice if the only chemical shifts of interest are

computed for other nuclei. You also remember that ECPs can deal with relativistic effects.

Relativistic effects are also an important consideration when predicting chemical shits. In terms of

computing absolute chemical shifts, they can be very large in heavy elements. For relative

chemical shifts the error is greatly reduced, because relativistic effects are primarily associated

with core orbitals, and core orbitals do not change much from one chemical environment to the

next. Nevertheless, accurate calculations involving atoms beyond the first row of transition metals

are challenging.

20.4. Performance of NMR Calculations

For molecules composed of only first-row atoms, heavy-atom chemical shifts can be

computed with a fair degree of accuracy. The following tables (from Cramer: Essentials of

11

Computational Chemistry) give you an idea what degree of accuracy you can expect from NMR

shielding (chemical shift) calculations.

Table 1. Absolute chemical shieldingsa

Note that proton (1H) chemical shifts are among the toughest to calculate precisely, because they

span a fairly narrow range - perhaps 15 ppm.

12

Even HF gives acceptable accuracy in most instances, although some improvements are

available in favorable instances from DFT (note, however, that LDA and B3LYP do particularly

badly – see the Table above) and MP2. The MP2 is quite accurate, but at relatively high cost in

terms of demand for computational resources. Various groups have demonstrated that errors from

levels having lower accuracy are sufficiently systematic that they may corrected by using empirical

factors, in the same spirit as the scaling of vibrational frequencies.

Table 2: Spin-spin coupling constants from LDA calculations and experiments

13

The calculation of spin–spin coupling is more difficult than that of a chemical shift, in part because

of the additional complications associated with two local magnetic moments, as opposed to one

moment and one external, uniform field. Moreover, the most commonly reported couplings in the

experimental literature are proton–proton couplings in organic and biological molecules, which

are again amongst the hardest to predict because they tend to be small in magnitude and the

absolute errors are correspondingly magnified when considered in a relative sense. Some

representative calculations are provided in the following table. Computed coupling constants are

quite sensitive to basis set, and accurate predictions require very flexible bases. As a rule, DFT is

much more robust than HF theory for predicting coupling constants, and the HF should be avoided

for this type of calculation.

20.4. Hyperfine Coupling Constants of Radicals

Molecules with unpaired electrons carry a non-zero electronic spin, which interacts with

the (non-zero) spins of the individual nuclei. The energy difference between the two the electronic

and nuclear spins being either aligned or opposed in the z-direction can be measured by electron

spin resonance (ESR) spectroscopy and defines the isotropic hyperfine splitting (h.f.s.) or hyperfine

coupling constant. To compute this quantity the molecular Hamiltonian is modified to include a

spin magnetic dipole at a particular nuclear position. The integral that results used to evaluate the

necessary perturbation is known as a Fermi contact integral.

For any open-shell molecule these coupling constants are calculated automatically as part

of the Population Analysis section labeled as “Fermi contact analysis”. These values

are given in atomic units and thus need to be converted into frequency, in this case as MHz. The

following expression accomplishes this:

bf = (16/3)(g/2)gIKBF (156)

where: g = 2.0023 observed free electron factor, K = 47.705336 MHz is a composite conversion

factor, BF is the atomic unit value of the hyperfine coupling calculated by Gaussian and gI is the

gyromagnetic ratio for a nucleus, which is the magnetic moment divided by the spin

Atom

Spin Magnetic Moment

Proton 1/2 2.792670

Carbon-13 1/2 0.7021

Nitrogen-14 1 0.4036

Make sure to include the keyword Density so that the population analysis uses the proper

electron density.

Calculation of hyperfine coupling requires that the localization of excess spin must be

accurately determined – at HF level the ROHF methodology is therefore not very useful, because

it cannot properly account for spin polarization. UHF on the other hand suffers from spin

contamination, which can lead to bad results. Projection (annihilation) of the spin contaminants is

14

usually and improvement. An important consideration is that the Fermi contact interaction (that’s

where “contact” comes from) arises from the electrons basis functions overlapping the nucleus.

Unfortunately, the core orbitals which have the highest overlap are usually treated in the most

approximate way, and the GTOs also have a wrong shape at the nucleus (the STO is the right

shape). For this reason, specific basis sets were developed for calculations of h.f.s. that correct for

this.

Overall, DFT is generally very good in computing h.f.s, except where delocalization is a

problem, which tends to occur in radicals. If that is the case, MP2 is usually the method that offers

the best price/performance ratio – though it is considerably more expensive than DFT.