This publication is intended to provide accurate and authoritative information on the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering professional services.
Preface to the First Edition
The lack of quality assessment is probably one of the reasons why computational chemistry has (had) a somewhat dismal reputation. This book grew out of a series of lecture notes that I used to teach a course in computational chemistry at Odense University, and the style of the book reflects its origins.
Preface to the Second Edition
A new Chapter 1 has been introduced, illustrating the similarities and differences between classical and quantum mechanics and providing some fundamental background. The number of alkanes that meet the above criteria is ~1010: clearly they will all have very similar and well-understood properties and there is no need to investigate all 1010 compounds.
1 Introduction
Fundamental Issues
The essence of the problem is that in many cases two-particle systems can be solved exactly by mathematical methods that produce solutions in the form of analytical functions. The numerically intensive tasks are typically related to simulating the behavior in the real world using a more or less sophisticated computational model.
Describing the System
Such data can be, for example, images generated by astronomical telescopes or gene sequences in the bioinformatics area to be compared. The main problem in such simulations is the multiscale nature of real problems, ranging from sub-nano to millimeters in spatial dimensions, and from femto- to milliseconds in the time domain.
Fundamental Forces
By solving the dynamic equation, the position and velocity of the particles can be predicted at later (or earlier) times relative to the initial conditions, i.e. only the electromagnetic interaction is important at the atomic and molecular level, and in the vast majority of cases simple Coulomb form (in atomic units) is sufficient:. 1.3) Within QED, the Coulomb interaction is only the zeroth order term, and the complete interaction can be written as an expansion in terms of the (inverse) speed of light, approx.
The Dynamical Equation
If the particle mass is constant, the derivative of the momentum p is the mass times the acceleration. In the limit of c→ ∞ the Dirac equation reduces to the Schrödinger equation, and the two.
Solving the Dynamical Equation
Separation of Variables
The time-dependent Schrödinger equation involves differentiation with respect to both time and position, the latter contained in the kinetic energy of the Hamiltonian operator. Inserting this into the time-dependent Schrödinger equation shows that the time and space variables of the wave function can be separated. 1.20).
Classical Mechanics
In this model, the orbit of one planet (eg Earth) is determined by taking into account the average interaction with all the other planets. The averaging effect is equivalent to spreading the mass of the other planets evenly along their orbits.
Quantum Mechanics
Since the Schrödinger equation is not completely separable in spherical polar coordinates, the constraints n>l≥ |m| exist. The n quantum number describes the size of the orbital, the quantum number describes the shape of the orbital, while the m quantum number describes the orientation of the orbital. the orbital relative to a fixed coordinate system. This argument could equally well be used for the second electron in relation to the first electron. The goal is therefore to calculate a set of self-consistent orbitals, and this can be done by iterative methods.
Chemistry
The goal of correlated methods for solving the Schrödinger equation is to calculate the residual correction due to the electron-electron interaction. Like an isolated atom, an atom in a molecule should consist of a nucleus and some electrons.
2 Force Field Methods
Introduction
Note that special atom types are defined for carbon atoms involved in small rings such as cyclopropane and cyclobutane. The idea of molecules being made up of atoms that are structurally similar in different molecules is implemented in models of force fields as types of atoms.
The Force Field Energy
Looking at the B—C bond, the torsion angle is defined as the angle formed by the A—B and C—D bonds as shown in Figure 2.7. The torsional energy is fundamentally different from Estrand Ebendin three aspects:. For polar molecules the (relative) conformational energies are therefore often of significantly lower accuracy than for non-polar systems. 2) The partial charge model gives a rather crude representation of the electrostatic potential around a molecule, with errors often in the 10–20 kJ/mol range.
Force Field Parameterization
In fact, the bond dissociation energy for a C-H bond depends on the environment:. the value for the aldehyde C—H bond in CH3CHO is 366 kJ/mol while it is 410 kJ/mol for C2H6.35 This can be roughly accounted for by assigning an average bond dissociation energy to a C—H bond , and a smaller correction based on larger structural units, such as CH3 and CHO groups. One problem with this procedure is that the atomic polarizability will naturally be modified by the bonding situation (i.e. the atom type), which is not taken into account by the Slater–Kirkwood equation.
Differences in Force Fields
Such force fields are, in principle, capable of describing molecules composed of elements of the entire periodic table, which have been designated as "all elements" in Table 2.4 below. Many force fields are undergoing development and the number of atom types is increasing as more and more systems are parameterized, so Table 2.4 can be seen as a "snapshot" of the situation when the data were collected.
Computational Considerations
In other words, the minimum energy geometry is insensitive to the exact value of the force constant. This is not the case for the other part of the unbound energy, the Coulomb interaction.
Validation of Force Fields
Validation of a force field is usually done by showing how accurately it reproduces the reference data that may or may not have been used in the actual parameterization. In this regard, it should be noted that the average error in the experimental data for hydrocarbons is 1.7 kJ/mol, i.e.
Practical Considerations
The result is that the set of parameters for a given force field is not constant in time and sometimes not even in geographic location. An obvious example is the MM2 force field, which exists in several different implementations that do not produce exactly the same results, but are nevertheless labeled as "MM2" results.
Advantages and Limitations of Force Field Methods
If the molecule is slightly out of the ordinary, it is very likely that there will be only poor quality parameters or none at all. The quality of the result can only be judged by comparing with other calculations on similar types of molecules for which relevant experimental data are available.
Transition Structure Modelling
This is difficult to evaluate accurately, as it is closely related to the accuracy of the force field used to describe the reactant and product structures. However, this requires that the force field is able to calculate relative energies of reactant and product, i.e.
Hybrid Force Field Electronic Structure Methods
The next level of refinement is called electron embedding, where atoms in the MM regions are allowed to polarize the QM region. A further enhancement, often called polarizing embedding, can be made by allowing the QM atoms to also polarize the MM region, ie.
3 Electronic Structure
The Adiabatic and Born–Oppenheimer Approximations
Without introducing any approximation, the total (exact) wave function can be written as an expansion in the full set of electronic functions, where the coefficients of the expansion are functions of the nuclear coordinates. 3.9) Nuclear kinetic energy is a sum of differential operators. The diagonal Born-Oppenheimer correction (DBOC) can be estimated relatively easily, since it is only the second derivative of the electronic wave function with respect to the nuclear coordinates, and is therefore closely related to the nuclear gradient and the second derivative of the energy (Section.
Self-Consistent Field Theory
The overall electronic wave function must be antisymmetric (change of sign) with respect to the exchange of two electron coordinates (since electrons are fermions with a spin of 1/2). The antisymmetry of the wave function can be achieved by building it from Slater determinants (SDs).
The Energy of a Slater Determinant
Note that the Jooperator involves "multiplication" with a matrix element with the same orbital on both sides, while the Kooperator "swaps" the two functions on the right-hand side of the g12operator. Using the complex conjugate properties in eq. the first two terms in eq. 3.39) must cancel, and the last two terms must cancel.
Koopmans’ Theorem
The orbital energies can be regarded as matrix elements of the Fock operator with the MOs (dropping the main notation and adopting the canonical orbitals). Eigenvalues corresponding to occupied orbitals are well defined and converge to a specific value as the size of the basis set increases.
The Basis Set Approximation
This is known as physical notation, where the order of the functions is given by the electron indices. The second part of the Fock matrix includes integrals over four basis functions and the gtwo electron operator.
An Alternative Formulation of the Variational Problem
Using the concepts from Chapter 16, the variational problem can be viewed as a rotation of the coordinate system. However, by performing a rotation of the coordinate system to the molecular orbitals, the matrix can be made diagonal, i.e.
Restricted and Unrestricted Hartree–Fock
The UHF wave function allows different spatial orbitals for the two electrons in an orbital. It has the disadvantage that the open-shell character is no longer present in the wave function; for example, it is not possible to calculate spin densities (i.e.
SCF Techniques
The number of two-electron integrals grows formally as the fourth power of the size of the basis set. Therefore, if the product of the density matrix elements and the upper limit of the integral is less than the cutoff, the integral does not need to be calculated.
Periodic Systems
The periodicity of the kernels in the system means that the square of the wave function must have the same periodicity. The price is that the solutions become a function of the reciprocal space vector k inside the first Brillouin zone.
Semi-Empirical Methods
The intermediate neglect differential overlap (INDO) approximation neglects all non-Coulomb-type two-center two-electron integrals in addition to those neglected by the NDDO approximations. The main difference between CNDO, INDO and NDDO is in the treatment of two-electron integrals.
Parameterization
The nucleus-nucleus repulsion of the MNDO model (Modified Neglect of Diatomic Overlap)47 has the form given in Eq. 3.93). Of the 12 new one-center two-electron integrals, only one (Gdd) is taken as a freely varied parameter.
Performance of Semi-Empirical Methods
However, it should be noted that the data in the tables refer to averages, so the order may be different for individual compounds or classes of compounds. It is clear that the PM3 method will perform better than AM1 in an average sense, since the two-electron integrals are optimized for a better fit to a given set of molecular data.
Hückel Theory
Since the diagonal elements depend only on the nature of the atom (i.e. the nuclear charge), this means for example that all carbon atoms have the same ability to attract electrons. Can be modified by the calculated atomic charge. 3.101) Wparameter determines the weight of the load on the diagonal elements.
Limitations and Advantages of Semi-Empirical Methods
The dependence on experimental data is not as severe as for force field methods, due to the more complex functional form of the model. The clear advantage of semiempirical methods over force field techniques is the ability to describe bond breaking and bond forming reactions.
4 Electron Correlation Methods
Excited Slater Determinants
The total number of determinants that can be generated depends on the size of the basis set: the larger the basis, the more virtual MOs, and the more excited. Just allowing excitations of the core electrons in a standard basis set does not "correlate" the core electrons.
Configuration Interaction
As shown in equations (4.8) and (4.9), the off-diagonal elements of the Fock matrix are CI matrix elements between the HF and singly excited states. Since the Hmatrix elements are essentially integrals of two electrons in the MO basis (Equation 4.8)), the iterative procedure can be formulated as integrally driven, i.e.
Illustrating how CI Accounts for Electron Correlation, and the RHF Dissociation Problem
The first two terms on the right have both electrons in the same nuclear center and describe the ionic contributions to the wave function, H+H−. The last two terms describe the covalent contributions to the H⋅H⋅ wave function. The HF wave function thus contains equal amounts of ionic and covalent contributions. The dissociation problem is solved in the case of the full CI wave function in this minimal basis.
The UHF Dissociation, and the Spin Contamination Problem
If the a and b bits are identical, there is no spin contamination and the UHF wavefunction is identical to RHF. By incorporating electronic correlation into the wave function, the UHF method introduces more biradical character into the wave function than RHF.
Size Consistency and Size Extensivity
Furthermore, the derivatives of the PUHF energy are not continuous at the RHF/UHF instability point. Projection of the wave function after added electron correlation, however, turns out to be a viable route.
Multi-Configuration Self-Consistent Field
This can be partially justified by the formula for the second-order perturbation energy correction (Section 4.8.1): the smaller the orbital energy difference, the larger the contribution to the correlation energy. If only part of the valence electrons are included in the active space, the CASSCF method tends to overestimate the importance of "biradical".
Multi-Reference Configuration Interaction
The correlation of the electrons in the C—H bonds, for example, will place more electron density on the carbon atoms. [4,4]-CASSCF also includes the two out-of-plane π orbitals in the active space, while [10,10]-CASSCF generates a full-valence CI wave function.
Many-Body Perturbation Theory
It is convenient to choose the perturbed wave function to be intermediately normalized, i.e. the overlap with the undisturbed wave function must be 1. If the reference wavefunction suffers from symmetry breaking (Section 3.8.3), the MP method is almost guaranteed. to give absurd results.
Coupled Cluster
Expanding the exponential in Eq. 4.54) and using the fact that the operator Hamiltonian contains only one- and two-electron operators (Eq. The energy calculated from these approximate odd and double amplitudes (Eq. 4.60)) will also be approximate.
Connections between Coupled Cluster, Configuration Interaction and Perturbation Theory
The triple amplitude can then be directly expressed in terms of single and double amplitudes, and MO integrals. The CCSD energy in Table 4.5 and the t4 amplitude in Table 4.6 indicate that the quadruple excited states in the CI wavefunction are mostly of the product type and not a true quadruple excited state.
Methods Involving the Interelectronic Distance
The coefficients in the CISDTQ and CCSDTQ wavefunctions (using intermediate normalization) for the dominant excitations are given in Table 4.6. The quadruply excited states enter the CI wavefunction with non-negligible weights, contributing ∼4% of the correlation energy (Table 4.5), but as shown in the last column of Table 4.6, these contributions are very well estimated by the product terms in the CC wave function.
