Hartree-Fock, approximation

The trial wave function in the Hartree-Fock approximation takes the form of a single Slater determinant  

Restricting the wave function by the form eq. (1.142) allows one to significantly reduce the calculation costs for all characteristics of a many-fermion system. Inserting eq. (1.142) into the energy expression (for the expectation value of the electronic Hamiltonian eq. (1.27)) and applying to it the variational principle with the additional condition of orthonormalization of the system of the occupied spin-orbitals 4 k (known in this context as molecular spin-orbitals) yields the system of integrodiffer-ential equations of the form (see e.g. [27])  

The Hartree-Fock equation eq. (1.143) can be rewritten using the Coulomb and exchange integral operators J and K, respectively  

If one-electron operators in eq. (1.143) are collected into a single operator defined according to  

With these notations the Hartree-Fock problem acquires the form of an eigen-value/eigenvector problem  

In order to show the reader how we calculate the dipole moment in practfce, let us use the Hartree-Fock approximation. Using the normalized Slater determinant t o) we have as the Hartree-Fock approximation to the dipole moment  

Oppenheimer approximation, the nuclei occupy some fixed positions in space. The electronic component of the dipole moment = (4 ol — according to the Slater-Condon rules (Appendix M on p. 986), amounts to = 

This is in principle all we can say about calculation of the dipole moment in the Hartree-Fock approximation. The rest belongs to the technical side. We choose a coordinate system and calculate all the integrals of type xk rxi), i-C- iXk xxi) iXk yxi)f ixk zxi)- The bond order matrix P is just a by-product of the Hartree-Fock procedure. 

One of the advantages of this method is that it breaks the many-electron Schrodinger equation into many simpler one-electron equations. Each one-electron equation is solved to yield a single-electron wave function, called an orbital, and an energy, called an orbital energy. The orbital describes the behavior of an electron in the net field of all the other electrons. 

The second approximation in HF calculations is due to the fact that the wave function must be described by some mathematical function, which is known exactly for only a few one-electron systems. The functions used most often are linear combinations of Gaussian-type orbitals exp(—nr ), abbreviated GTO. The wave function is formed from linear combinations of atomic orbitals or, stated more correctly, from linear combinations of basis functions. Because of this approximation, most HF calculations give a computed energy greater than the Hartree-Fock limit. The exact set of basis functions used is often specified by an abbreviation, such as STO—3G or 6—311++g. Basis sets are discussed further in Chapters 10 and 28. 

The Gaussian functions are multiplied by an angular function in order to give the orbital the symmetry of a s, p, d, and so on. A constant angular term yields s symmetry. Angular terms of x, y, z give p symmetry. Angular terms of xy, xz, yz, x —y, Az —2x —2y yield d symmetry. This pattern can be continued for the other orbitals. 

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

A variation on the HF procedure is the way that orbitals are constructed to reflect paired or unpaired electrons. If the molecule has a singlet spin, then the same orbital spatial function can be used for both the a and P spin electrons in each pair. This is called the restricted Hartree-Fock method (RHF). 

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Although it is now somewhat dated, this book provides one of the best treatments of the Hartree-Fock approximation and the basic ideas involved in evaluating the correlation energy. An especially valuable feature of this book is that much attention is given to how these methods are actually implemented. 

Configuration interaction (Cl) is a systematic procedure for going beyond the Hartree-Fock approximation. A different systematic approach for finding the correlation energy is perturbation theory... 

Another distinguishing aspect of MO methods is the extent to which they deal with electron correlation. The Hartree-Fock approximation does not deal with correlation between individual electrons, and the results are expected to be in error because of this, giving energies above the exact energy. MO methods that include electron correlation have been developed. The calculations are usually done using MoUer-Plesset perturbation theoiy and are designated MP calculations." ... 

In addition, if one goes beyond the Hartree-Fock approximation to something like the configuration interaction approach there is an important sense in which one has gone beyond the picture of a certain number of electrons into a set of orbitals.10 If one insists on picturing this, then rather than just every electron being in eveiy possible orbital... 

It is presumed that this function is optimized in the Hartree-Fock approximation. 

Before discussing the correlation error, we will make some introductory remarks about the Hartree-Fock approximation based on the use of the Slater determinant (Eq. 11.38). We note that, if we... 

These relations show that the Fock-Dirac density matrix is identical with the first-order density matrix, and that consequently the first-order density matrix determines all higher-order density matrices and then also the entire physical situation. This theorem is characteristic for the Hartree-Fock approximation. 

The second-order density matrix is in the Hartree-Fock approximation given by Eqs. 11.44 and 11.53, and we obtain directly... 

The correlation error can, of course, be defined with reference to the Hartree scheme but, in modem literature on electronic systems, one usually starts out from the Hartree-Fock approximation. This means that the main error is due to the neglect of the Coulomb correlation between electrons with opposite spins and, unfor-tunetely, we can expect this correlation error to be fairly large, since we force pairs of electrons with antiparallel spins together in the same orbital in space. The background for this pairing of the electrons is partly the classical formulation of the Pauli principle, partly the mathematical fact that a single determinant in such a case can... 

We note that the virial theorem is automatically fulfilled in the Hartree-Fock approximation. This result follows from the fact that the single Slater determinant (Eq. 11.38) built up from the Hartree-Fock functions pk x) satisfying Eq. 11.46 is the optimum wave function of this particular form, and, since this wave function cannot be further improved by scaling, the virial theorem must be fulfilled from the very beginning. If we consider a stationary state with the nuclei in their equilibrium positions, we have particularly Thf = — Fhf, and for the correlation terms follows consequently that... 

According to Eq. 11.67, the correlation energy is simply defined as the difference between the exact energy and the energy of the Hartree-Fock approximation. Let us repeat this definition in a more precise form ... 

The correlation energy for a certain state with respect to a specified Hamiltonian is the difference between the exact eigenvalue of the Hamiltonian and its expectation value in the Hartree-Fock approximation for the state under consideration. 

We see immediately the connection with the one-electron scheme, but we note that the emphasis is here on the word "complete, whereas, in the Hartree-Fock approximation, one is looking for a finite set of best spin orbitals. 

In the bibliography, we have tried to concentrate the interest on contributions going beyond the Hartree-Fock approximation, and papers on the self-consistent field method itself have therefore not been included, unless they have also been of value from a more general point of view. However, in our treatment of the correlation effects, the Hartree-Fock scheme represents the natural basic level for study of the further improvements, and it is therefore valuable to make references to this approximation easily available. For atoms, there has been an excellent survey given by Hartree, and, for solid-state, we would like to refer to some recent reviews. For molecules, there does not seem to exist something similar so, in a special list, we have tried to report at least the most important papers on molecular applications of the Hartree-Fock scheme, t... 

Husimi, K., Proc. Phys.-Math. Soc. Japan 22, 264, "Some formal properties of the density matrix." Introduction of the concept of reduced density matrix. Statistical-mechanical treatment of the Hartree-Fock approximation at an arbitrary temperature and an alternative method of obtaining the reduced density matrices are discussed. 

Lowdin, P.-O., Phys. Rev. 97, 1474, 1490, 1509, Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals and convergence problems in the method of configuration interaction. II. Study of the ordinary Hartree-Fock approximation. III. Extension of the Har-tree-Fock scheme to include degenerate systems and correlation effects. ... 

Lagrangian as a functional -(v,v). Note however that unlike functionals used in the Time-dependent Hartree Fock approximation (14), this Lagrangian is not complex analytic in the variables (v,v) separately. 

To properly describe electronic rearrangement and its dependence on both nuclear positions and velocities, it is necessary to develop a time-dependent theory of the electronic dynamics in molecular systems. A very useful approximation in this regard is the time-dependent Hartree-Fock approximation (34). Its combination with the eikonal treatment has been called the Eik/TDHF approximation, and has been implemented for ion-atom collisions.(21, 35-37) Approximations can be systematically developed from time-dependent variational principles.(38-41) These can be stated for wavefunctions and lead to differential equations for time-dependent parameters present in trial wavefunctions. 

The eikonal/time-dependent Hartree Fock approximation and extensions. 

A number of techniques have been introduced since 1930 to overcome the problem of the neglect of correlation energy in the Hartree-Fock approximation. 

The most famous case concerns the symmetry breaking in the Hartree-Fock approximation. The phenomenon appeared on elementary problems, such as H2, when the so-called unrestricted Hartree-Fock algorithms were tried. The unrestricted Hartree-Fock formalism, using different orbitals for a and p electrons, was first proposed by G. Berthier [5] in 1954 (and immediately after J.A. Pople [6] ) for problems where the number of a andp electrons were different. This formulation takes the freedom to deviate from the constraints of being an eigenfunction. 

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