## Many Body Perturbation Theory

The idea in perturbation methods is that the problem at hand only differs slightly from a problem that has already been solved (exactly or approximately).The solution to the given problem should therefore in some sense be close to the solution to the already known system. This is described mathematically by defining a Hamiltonian operator that consists of two parts, a reference (H0) and a perturbation (H'). The premise of perturbation methods is that the H' operator in some sense is small...

## F i F

F ( h f a) + k g f f k) - f f g f 0 When a perturbation is introduced, the stationary condition means that the orbitals must change, which may be described as a mixing of the unperturbed MOs. In other words, the stationary orbitals in the presence of a perturbation are given by a unitary transformation of the unperturbed orbitals (see also Section 3.6). The U matrix describes how the MOs change, i.e. it contains the derivatives of the MO coefficients. In the absence of a perturbation, U is the...

## Reduced scaling techniques

The computational bottleneck in HF methods is the calculation of the two-electron Coulomb and exchange terms arising from the electron-electron repulsion. In non-metallic systems, the exchange term is quite short-ranged, while the Coulomb interac tion is long-ranged. In the large system limit, the Coulomb integrals thus dominate the computational cost. By using the screening techniques described in the previous section, the scaling in the large system limit will eventually be reduced from...

## The Basis Set Approximation

For small highly symmetric systems, such as atoms and diatomic molecules, the Hartree-Fock equations may be solved by mapping the orbitals on a set of grid points, and these are referred to as numerical Hartree-Fock methods.8 However, essentially all calculations use a basis set expansion to express the unknown MOs in terms of a set of known functions. Any type of basis functions may in principle be used exponential, Gaussian, polynomial, cube functions, wavelets, plane waves, etc. There are...

## CI Matrix elements

The CI matrix elements Hij can be evaluated by the strategy employed for calculating the energy of a single determinant used for deriving the Hartree-Fock equations (Section 3.3). This involves expanding the determinants in a sum of products of MOs, thereby making it possible to express the CI matrix elements in terms of MO integrals. There are, however, some general features that make many of the CI matrix elements equal to zero. The Hamiltonian operator (eq. (3.23)) does not contain spin,...

## Complete Neglect of Differential Overlap Approximation CNDO

In the Complete Neglect of Differential Overlap (CNDO) approximation all the Coulomb two-electron integrals are subjected to the condition in eq. (3.87), including the one-centre integrals, and are again parameterized as in eq. (3.88). The approximations for the one-electron integrals in CNDO are the same as for INDO. The Pariser-Pople-Parr (PPP) method can be considered as a CNDO approximation where only n-electrons are treated. The main difference between CNDO, INDO and NDDO is in the...

## Choice of Coordinates

Naively one may think that any set of coordinates that uniquely describes the function is equally good for optimization. This is not the case A good set of coordinates may transform a divergent optimization into a convergent one, or increase the rate of convergence. We will look specifically at the problem of optimizing a geometry given an energy function depending on nuclear coordinates, but the same considerations hold equally well for other types of optimization. We will furthermore use the...

## Change of Coordinate System

In many cases, it is possible to simplify a problem by choosing a particular coordinate system. It is therefore important to be able to describe how vectors and matrices change when switching from one coordinate system to another. Some coordinate transformations are non-linear, such as converting from a Cartesian to a spherical polar system. Here the r, 0, j coordinates are related to the x, y, z coordinates by square root and trigonometric functions, as shown in Figure 16.1. Other coordinate...

## Force Field Parameterization

Having settled on the functional description and a suitable number of cross terms, the problem of assigning numerical values to the parameters arises. This is by no means trivial.36 Consider for example MM2(91) with 71 atom types. Not all of these can form stable bonds with each other, hydrogens and halogens can only have one bond, etc. For the sake of argument, however, assume that the effective number of atom types capable of forming bonds between each other is 30. Each of the 71 atom types...

## Molecular Docking

An important example of a global optimization problem is determining the best alignment of two molecules with respect to each other, typically trying to fit a small molecule into a large protein structure, a process called docking. Given an X-ray structure of an enzyme, preferably with a bound ligand to identify the active site, the ligand can be removed, and other (virtual) compounds may be docked into the active site to possibly identify new molecules with a stronger binding affinity. Since...

## Even and Well Tempered Basis Sets

The optimization of basis function exponents is an example of a highly non-linear optimization problem (Chapter 12). When the basis set becomes large, the optimization problem is no longer easy. The basis functions start to become linearly dependent (the basis set approaches completeness) and the energy becomes a very flat function of the exponents. Analyses of basis sets that have been optimized by variational methods reveal that the ratio between two successive exponents is approximately...

## Monte Carlo Methods

On of the advantages of Monte Carlo methods is the ease with which they can be implemented in computer programs. The heart of the algorithm is a random number generator, and the ability to calculate the energy of the system for a given set of coordinates. Although truly random numbers are difficult to come by, several implementations of pseudo-random number generators are available. A pseudo-random number generator indicates a computer implementation of an algorithm that produces a sequence of...

## Slater and Gaussian Type Orbitals

There are two types of basis functions (also called Atomic Orbitals (AO), although they in general are not solutions to an atomic Schr dinger equation) commonly used Introduction to Computational Chemistry, Second Edition. Frank Jensen. 2007 John Wiley & Sons, Ltd in electronic structure calculations Slater Type Orbitals (STO) and Gaussian Type Orbitals (GTO). Slater type orbitals2 have the functional form shown in eq. (5.1). Xc,n,im (r, e, j) NYm (0, j )rn -1e-Zr (5.1) Here N is a...

## Exchange and Correlation Holes

We now return to the problem of expressing the exchange-correlation energy as a functional of p ( p1). Since the exchange energy is by far the largest contributor to Exc (cf. the values for the neon atom in Section 6.2), one may reasonably ask why not calculate this term exactly from orbitals (analogous to the kinetic energy), by the formula known from wave mechanics (eq. (3.31)), and only calculate the computationally difficult part, the correlation energy, by DFT. Although this has been...

## M0llerPlesset perturbation theory

So far, the theory has been completely general. In order to apply perturbation theory to the calculation of correlation energy, the unperturbed Hamiltonian operator must be selected. The most common choice is to take this as a sum over Fock operators, leading to M0ller-Plesset (MP) perturbation theory.9 The sum of Fock operators counts the (average) electron-electron repulsion twice (eq. (3.44)), and the perturbation becomes the exact Vee operator minus twice the < yee) operator. The operator...

## Qualitative Molecular Orbital Theory

Frontier molecular orbital theory is closely related to various schemes of qualitative orbital theory where interactions between fragment MOs are considered.14 Ligand field theory, as commonly used in systems involving coordination to metal atoms, can be considered as a special case where only the d-orbitals on the metal and selected orbitals of the ligands are considered. Two interacting orbitals will in general produce two new orbitals, having lower and higher energies than the...

## Plane Wave Basis Functions

Rather than starting with basis functions aimed at modelling the atomic orbitals (STOs or GTOs), and forming linear combination of these to describe orbitals for the whole system, one may use functions aimed directly at the full system. For modelling extended (infinite) systems, for example a unit cell with periodic boundary conditions, this suggests the use of functions with an infinite range. The outer valence electrons in metals behave almost like free electrons, which leads to the idea of...

## Final Considerations

Should DFT methods be considered ab initio or semi-empirical If ab initio is taken to mean the absence of fitting parameters, LSDA methods are ab initio but gradient-corrected methods may or may not be. The LSDA exchange energy contains no parameters and the correlation functional is known accurately as a tabulated function of the density. The use of a parameterized interpolation formula in practical calculations does not represent fitting in order to improve the performance for atomic and...

## Conformational Sampling and the Global Minimum Problem

The methods described in Section 12.2 can only locate the nearest minimum, which is normally a local minimum, when starting from a given set of variables. In some cases, the interest is in the lowest of all such minima, the global minimum in other cases it is important to sample a large (preferably representative) set of local minima. Considering that the number of minima typically grows exponentially with the number of variables, the global optimization problem is an extremely difficult task...

## Initial guess orbitals

The quality of the initial guess orbitals influences the number of iterations necessary for achieving convergence. As each iteration involves a computational effort proportional to Mbasis, it is of course desirable to generate as good a guess as possible. Different start orbitals may in some cases result in convergence to different SCF solutions, or make the difference between convergence and divergence. One possible way of generating a set of start orbitals is to diagonalize the Fock matrix...

## Wave Function Analysis

The previous chapters have focused on various methods for obtaining more or less accurate solutions to the Schrodinger equation. The natural byproduct of determining the electronic wave function is the energy. However, there are many other properties that may be derived. Although the quantum mechanical description of a molecule is in terms of positive nuclei surrounded by a cloud of negative electrons, chemistry is still formulated as atoms held together by bonds.This raises questions such as...

## Connections between Coupled Cluster Configuration Interaction and Perturbation Theory

The general cluster operator is given by eq. (4.69), where terms have been collected according to the excitation they generate. e T 1+Ti + (T2 + 2 T2 ) + (Ta + T2T1 +1T13 ) + (T4 + T3 Ti + i T22 + T2 Ti2 + Ti4 ) + Each of the operators in a given parenthesis generates all the excited determinants of the given type. Both T2 and T2 generate all doubly excited determinants, and the terms in eq. (4.69) generate all determinants that are included in a CISDTQ calculation. The cluster expansion can be...

## Electron Correlation Methods

The Hartree-Fock method generates solutions to the Schrodinger equation where the real electron-electron interaction is replaced by an average interaction (Chapter 3). In a sufficiently large basis set, the HF wave function is able to account for 99 of the total energy, but the remaining 1 is often very important for describing chemical phenomena. The difference in energy between the HF and the lowest possible energy in the given basis set is called the Electron Correlation (EC) energy.1...

## Multi Configuration Self Consistent Field

The Multi-Configuration Self-Consistent Field (MCSCF) method can be considered as a CI where not only are the coefficients in front of the determinants (eq. (4.2)) optimized by the variational principle, but the MOs used for constructing the determinants are also optimized.6 The MCSCF optimization is iterative like the SCF procedure (if the multi configuration is only one, it is simply HF). Since the number of MCSCF iterations required for achieving convergence tends to increase with the number...

## Poisson Boltzmann methods

The Poisson equation is a second-order differential equation describing the connection between the electrostatic potential f, the charge distribution p and the dielectric constant e.64 Note that the dielectric constant may depend on the position. When it is independent of the position (i.e. truly a constant), eq. (14.52) becomes eq. (14.53). Gcavity + AGdispersion g SAS + Gcavity + AGdispersion g SAS + If the charge distribution is a point charge, the solution of eq. (14.53) reduces to the...

## Time Dependent Methods

At a finite temperature, the average kinetic energy is directly related to the temperature and the molecule(s) explores a part of the surface with energies lower than the typical kinetic energy. One possible way of simulating the behaviour at a finite temperature is by allowing the system to evolve according to the relevant dynamical equation (Section 1.4). For nuclei, this is normally Newton's second law, although the (nuclear) Schr dinger equation must be used for including quantum effects,...

## Composite Extrapolation Procedures

In principle, the large majority of systems can be calculated with a high accuracy by using a highly correlated method such as CCSD(T) and performing a series of calculations with systematically larger basis sets in order to extrapolate to the basis set limit. In practice, even a single water molecule is demanding to treat in this fashion (Chapter 11).Various approximate procedures have therefore been developed for estimating the infinite correlation, infinite basis limit (Figure 4.3) as...

## Intrinsic Reaction Coordinate Methods

The optimization methods described in Sections 12.2-12.4 concentrate on locating stationary points on an energy surface. The important points for discussing chemical reactions are minima, corresponding to reactant(s) and product(s), and saddle points, corresponding to transition structures. Once a TS has been located, it should be verified that it indeed connects the desired minima. At the TS the vibrational normal coordinate associated with the imaginary frequency is the reaction coordinate...

## Newton Raphson methods

The Newton-Raphson (NR) method expands the true function to second order around the current point x0. f(x) f (x 0) + g t(x - xo) + i (x - xo )t H(x - xo) (12.9) Requiring the gradient of the second-order approximation in eq. (12.9) to be zero produces the step in eq. (12.10). In the coordinate system (x') where the Hessian is diagonal (i.e. performing a unitary transformation, see Section 16.2), the NR step may be written as in eq. (12.11). Here fi is the projection of the gradient along the...

## Electronic Structure Methods Independent Particle Models

If we are interested in describing the electron distribution in detail, there is no substitute for quantum mechanics. Electrons are very light particles and they cannot be described correctly even qualitatively by classical mechanics. We will in this chapter and in Chapter 4 concentrate on solving the time-independent Schr dinger equation, which in shorthand operator form is given in eq. (3.1). If solutions are generated without reference to experimental data, the methods are usually called ab...

## Contracted Basis Sets

One disadvantage of all energy-optimized basis sets is the fact that they primarily depend on the wave function in the region of the inner-shell electrons. The 1s-electrons account for a large part of the total energy, and minimizing the energy will tend to make the basis set optimum for the core electrons, and less so for the valence electrons. However, chemistry is mainly dependent on the valence electrons. Furthermore, many properties (for example polarizability) depend mainly on the wave...

## More OFerrall Jencks Diagrams

The BEP Hammond Marcus treatment only considers changes due to energy differences between the reactant and product, i.e. changes in the TS position along the reaction coordinate. It is often useful also to include changes that may occur in a direction perpendicular to the reaction coordinate. Such two-dimensional diagrams are associated with the names of More O'Ferrall and Jencks (MOJ diagrams).34 Consider for example the Cope rearrangement of 1,5-hexadiene. Since the reaction is degenerate the...

## Effective Core Potentials

Systems involving elements from the lower part of the periodic table have a large number of core electrons. These are, as already mentioned, unimportant in a chemical sense, but it is necessary to use a large number of basis functions to expand the corresponding orbitals, otherwise the valence orbitals will not be properly described (due to a poor description of the electron-electron repulsion). In the lower half of the periodic table relativistic effects further complicate matters (see Chapter...

## Quantum Monte Carlo Methods

Monte Carlo methods refer to techniques for obtaining the value of a multi-dimensional integral of a function by randomly probing its value within the whole variable space, and estimating the integral by statistical averaging. In the limit of an infinite number of sampling points, the result is identical to that obtained from an analytical integration, but for a finite number of points, the calculated value is given as an average with an associated standard deviation. The standard deviation,...

## Woodward Hoffmann Rules

The Woodward-Hoffmann (W-H) rules are qualitative statements regarding relative activation energies for two possible modes of reaction, which may have different stere-ochemical outcomes.18 For simple systems, the rules may be derived from a conservation of orbital symmetry, but they may also be generalized by an FMO treatment with conservation of bonding. Let us illustrate the Woodward-Hoffmann rules with a couple of examples, the preference of the 4 + 2 over the 2 + 2 product for the reaction...

## Classification of Basis Sets

Having decided on the type of function (STO GTO) and the location (nuclei), the most important factor is the number of functions to be used. The smallest number of functions possible is a minimum basis set. Only enough functions are employed to contain all the electrons of the neutral atom(s). For hydrogen (and helium) this means a single s-function. For the first row in the periodic system it means two s-functions (1s and 2s) and one set of p-functions (2px, 2py and 2pz). Lithium and beryllium...

## Frontier Molecular Orbital Theory

Frontier Molecular Orbital FMO theory attempts to predict relative reactivity based on properties of the reactants. It is commonly formulated in term of perturbation theory, where the energy change in the initial stage of a reaction is estimated and extrapolated to the transition state.1 For a reaction where two different modes of reaction are possible, this may be illustrated as shown in Figure 15.1. The reaction mode that involves the least energy change in the initial stage is assumed also...

## Pseudospectral Methods

The goal of pseudospectral methods76 is to reduce the formal M4 dependence of the Coulomb and exchange operators in the basis set representation two-electron integrals, eq. 3.52 to M3. This can be accomplished by switching between a grid representation in the physical space the three-dimensional Cartesian space and the spectral representation in the function space the basis set . Consider the following Coulomb contribution to the Fa element of the Fock matrix eq. 3.52 similar considerations...

## Small rings and conjugated systems

It has already been mentioned that small rings present a problem as their equilibrium angles are very different from their acyclic cousins. One way of alleviating this problem is to assign new atom types. If a sufficient number of cross terms is included, however, the necessary number of atom types can actually be reduced. Some force fields have only one sp3-carbon atom type, covering bonding situations from cyclopropane to linear alkanes with the same set of parameters. The necessary...