Population analyses

The Mulliken population analysis is a simple way of gaining some useful information about the distribution of the electrons in the molecule. Let us assume again a UHF wave function  

In equation (A.77) is defined the atomic orbital population P,-. Summing all of the P, that belong to the same atom, I, yields the atomic population / [equation (A.78)]. The net charge qj on atom I is just the difference between the nuclear charge Zi and the atomic population, 

A widely used (and widely criticized) method to analyze SCF wave functions is population analysis, introduced by Mulliken. He proposed a method that apportions the electrons of an n-electron molecule into net populations in the basis functions Xr and overlap populations for all possible pairs of basis functions. 

Integrating this equation over three-dimensional space and using the fact that and the A i s are normalized, we get 

By summing over the occupied MOs, we obtain the Mulliken net population in Xr and the overlap population for the pair Xr and Xs as 

The sum of all the net and overlap populations equals the total number of electrons in the molecule (Problem 15.20) = ti. 

EXAMPLE For the H2O MOs in (15.20), calculate the net and overlap population contributions from the 2ai MO and find n, for each basis function. Use Hjls and H2IS as basis functions, rather than the symmetry-adapted basis functions. 

Chemists are able to do research much more efficiently if they have a model for understanding chemistry. Population analysis is a mathematical way of partitioning a wave function or electron density into charges on the nuclei, bond orders, and other related information. These are probably the most widely used results that are not experimentally observable. 

Knowledge of the molecular wavefunction enables us to determine the electron density at any given point in space. Here we inquire about the amount of electronic charge that can be associated in a meaningful way with each individual atom of a A -electron system. Our analysis covers Mulliken s celebrated population analysis [31], as well as a similar, closely related method. 

In MO calculations, the total electron density is represented as the sum of all populated MOs. The electron density at any atom can be obtained by summing the electron density associated with the basis set orbitals for each atom. Electron density shared by two or more atoms, as indicated by the overlap integral, is partitioned equally among them. This is called a Mulliken population analysis  

Schleyer, and J. A. Pople, Ab Initio Molecular Orbital Theory, Wiley-Interscience, New York, 1986, pp. 118-121 A. Szabo and N. S. Ostlund, Modem Quantum Chemistry Introduction to Advanced Electronic Structure Theory, Macmillan, New York, 1982. 

Once an approximation to the wavefunction of a molecule has been found, it can be used to calculate the probable result of many physical measurements and hence to predict properties such as a molecular hexadecapole moment or the electric field gradient at a quadrupolar nucleus. For many workers in the field, this is the primary objective for performing quantum-mechanical calculations. But from 

Secondly, making measurements on atomic and molecular systems generally interferes with the system. If we were to repeatedly make measurements on a single system, we would change the system at each measurement and so would not be dealing necessarily with the same system. The correct interpretation is a statistical one. We would have to prepare a very large number of systems all in the same electronic state, and then do the measurements on all of them. 

For a preliminary survey of the electron density, it is usual to make a pictorial representation as we did in previous chapters. Whilst such diagrams do not carry much information, they do provide a theoretical measure which can be compared to the results of X-ray diffraction studies. A whole volume of the Transactions of the American Chemical Society (1972) was devoted to the Symposium Experimental and Theoretical Studies of Electron Densities . 

For the purposes of a purely theoretical analysis of molecular electronic structure, we need more detailed information. The term population analysis was introduced in a series of papers by Mulliken in 1955, but the basic ideas had already been anticipated by Mulliken himself, and by other authors. The technique has been widely applied since Mulliken s 1955 papers, because it is very simple and has the apparent virtue of being quantitative . The word quantitative seems to mean two different things to different authors  

Most users of population analysis seem to be concerned with the first meaning. Take the LCAO-MO treatment of dihydrogen as an example. We focus on the electron density 

Atomic partial charges are a difficult concept in quantum chemistry. On the one hand, assigning charges to individual atoms in a molecule is very close to the classical interpreta- 

Within the harmonic oscillator approximation, the ZPVE is obtained as 

The eigenvector associated with the diagonal mass-weighted Hessian defines the atomic motion associated with that particular frequency. The vibrational frequencies can also be used to compute the entropy of the molecule and ultimately the Gibbs free energy.  

Since there is no operator that produces the atomic population, it is not an observable and so the procedure for computing N(k) is arbitrary. There are two classes of methods for computing the atomic population those based on the orbital population and those based on a spatial distribution.  

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Mulliken population analysis is a trivial calculation to perform once a self-consistent field has been established and the elements of the density matrix have been determined. 

In the Lowdin approach to population analysis [Ldwdin 1970 Cusachs and Politzer 1968] the atomic orbitals are transformed to an orthogonal set, along with the molecular orbital coefficients. The transformed orbitals in the orthogonal set are given by ... 

Ldwdin population analysis avoids the problem of negative populations or populations greater than 2. Some quantum chemists prefer the Ldwdin approach to that of Mulliken as the charges are often closer to chemically intuitive values and are less sensitive to basis set. 

Journal of the American Chemical Society 112 114-118. iiai ii rach S M 1994. Population Analysis and Electron Densities from Quantum Mechanics. In Lipkowitz K B and D B Boyd (Editors). Reviews in Computational Chemistry Volume 5. New York, VCl 1 Publishers, pp 171-227. 

The Fenske-Hall method is a modification of crystal held theory. This is done by using a population analysis scheme, then replacing orbital interactions with point charge interactions. This has been designed for the description of inorganic metal-ligand systems. There are both parameterized and unparameterized forms of this method. 

There is some ambiguity about Mulliken population analysis in the literature. This is because various software packages print different portions of the analysis and may name them slightly differently. The description here follows some of the more common conventions. 

The Lowdin population analysis scheme was created to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme, which it does. It is different in that the atomic orbitals are first transformed into an orthogonal set, and the molecular orbital coefficients are transformed to give the representation of the wave function in this new basis. This is less often used since it requires more computational work to complete the orthogonalization and has been incorporated into fewer software packages. The results are still basis-set-dependent. 

Natural bond order analysis (NBO) is the name of a whole set of analysis techniques. One of these is the natural population analysis (NPA) for obtaining... 

This results in a population analysis scheme that is less basis set dependent than the Mulliken scheme. Flowever, basis set effects are still readily apparent. This is also a popular technique because it is available in many software packages and researchers find it convenient to use a method that classifies the type of orbital. 

A much less basis set dependent method is to analyze the total electron density. This is called the atoms in molecules (AIM) method. It is designed to examine the small effects due to bonding in the primarily featureless electron density. This is done by examining the gradient and Laplacian of electron density. AIM analysis incorporates a number of graphic analysis techniques as well as population analysis. The population analysis will be discussed here and the graphic techniques in the next chapter. 

Recent progress in this field has been made in predicting individual atoms contribution to optical activity. This is done using a wave-functioning, partitioning technique roughly analogous to Mulliken population analysis. 

Vector quantities, such as a magnetic field or the gradient of electron density, can be plotted as a series of arrows. Another technique is to create an animation showing how the path is followed by a hypothetical test particle. A third technique is to show flow lines, which are the path of steepest descent starting from one point. The flow lines from the bond critical points are used to partition regions of the molecule in the AIM population analysis scheme. 

The subscripts i and j denote two nuclei one in the QM region and one in the MM region. The atomic charges for the MM atoms are obtained by any of the techniques commonly used in MM calculations. The atomic charges for the QM atoms can be obtained by a population analysis scheme. Alternatively, there might be a sum of interactions with the QM nuclear charges plus the interaction with the electron density, which is an integral over the electron density. 

Unrestricted calculations often incorporate a spin annihilation step, which removes a large percentage of the spin contamination from the wave function. This helps minimize spin contamination but does not completely prevent it. The final value of (,S y is always the best check on the amount of spin contamination present. In the Gaussian program, the option iop(5/14=2) tells the program to use the annihilated wave function to produce the population analysis. 

The consistent total energy makes it possible to compute singlet-triplet gaps using RHF for the singlet and the half-electron calculation for the triplet. Koopman s theorem is not followed for half-electron calculations. Also, no spin densities can be obtained. The Mulliken population analysis is usually fairly reasonable. 

Molecular descriptors must then be computed. Any numerical value that describes the molecule could be used. Many descriptors are obtained from molecular mechanics or semiempirical calculations. Energies, population analysis, and vibrational frequency analysis with its associated thermodynamic quantities are often obtained this way. Ah initio results can be used reliably, but are often avoided due to the large amount of computation necessary. The largest percentage of descriptors are easily determined values, such as molecular weights, topological indexes, moments of inertia, and so on. Table 30.1 lists some of the descriptors that have been found to be useful in previous studies. These are discussed in more detail in the review articles listed in the bibliography. 

Many semiempirical methods have been created for modeling organic compounds. These methods correctly predict many aspects of electronic structure, such as aromaticity. Furthermore, these orbital-based methods give additional information about the compounds, such as population analysis. There are also good techniques for including solvation elfects in some semiempirical calculations. Semiempirical methods are discussed further in Chapter 4. 

Population analysis poses a particularly difficult problem for the / block elements. This is because of the many possible orbital combinations when both /and d orbitals are occupied in the valence. Although programs will generate a population analysis, extracting meaningful information from it can be very difficult. 

The ah initio module can run HF, MP2 (single point), and CIS calculations. A number of common basis sets are included. Some results, such as population analysis, are only written to the log file. One test calculation failed to achieve SCF convergence, but no messages indicating that fact were given. Thus, it is advisable to examine the iteration energies in the log file. 

A number of molecular properties can be computed. These include ESR and NMR simulations. Hyperpolarizabilities and Raman intensities are computed using the TDDFT method. The population analysis algorithm breaks down the wave function by molecular fragments. IR intensities can be computed along with frequency calculations. 

Crystal can compute a number of properties, such as Mulliken population analysis, electron density, multipoles. X-ray structure factors, electrostatic potential, band structures, Fermi contact densities, hyperfine tensors, DOS, electron momentum distribution, and Compton profiles. 

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