Hartree-Fock-Roothaan procedure

Combining the LCAO approximation for the MOs with the HE method led Roothaan to develop a procedure to obtain the SCE solutions. We will discuss here only the simplest case where all MOs are doubly occupied with one electron that is spin up and one that is spin down, also known as a closed-shell wavefiinction. The open-shell case is a simple extension of these ideas. The procedure rests upon transforming the set of equations listed in Eq. (1.7) into matrix form 

The expression in Eq. (1.12) is for a closed-shell wavefunction, but it can be defined for a more general wavefunction by analogy. 

The matrix approach is advantageous because a simple algorithm can be established for solving Eq. (1.10). First, a matrix X is found which transforms the normalized AOs Xu into the orthonormal set x n 

The Hartree-Fock-Roothaan algorithm is implemented by the following steps. 

One last point concerns the nature of the MOs that are produced in this procedure. These orbitals are such that the energy matrix e will be diagonal, with the diagonal elements being interpreted as the MO energy. These MOs are referred to as the canonical orbitals. One must be aware that all that makes them unique is 

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This restriction is not demanded. It is a simple way to satisfy the Pauli exclusion principle, but it is not the only means for doing so. In an unrestricted wavefunction, the spin-up electron and its spin-down partner do not have the same spatial description. The Hartree-Fock-Roothaan procedure is slightly modified to handle this case by creating a set of equations for the a electrons and another set for the p electrons, and then an algorithm similar to that described above is implemented. 

In order to solve for the energy and wavefunction within the Hartree-Fock-Roothaan procedure, the AOs must be specified. If the set of AOs is infinite, then the variational principle tells us that we will obtain the lowest possible energy within the HF-SCF method. This is called the HF limit, This is not the actual energy of the molecule recall that the HF method neglects instantaneous electron-electron interactions, otherwise known as electron correlation. 

When the Schrodinger equation is solved in the Hartree-Fock— Roothaan procedure, the coefficients c,> are obtained and the wavefunction is at hand.2 Unfortunately, all the chemical information is contained in this wave-function, and it is expressed as a (very) long list of coefficients. As an example, a restricted Hartree-Fock calculation of benzene using the 6-31G basis set will have 102 atomic orbitals and 21 doubly occupied MOs for a total of 2142 coefficients. For the chemist, the interesting and pertinent data are entangled in a series of numbers, and the question becomes how to extract the chemical concepts from these numbers. 

With radicals there is no convenient method like the Hartree-Fock-Roothaan procedure commonly used for closed-shell systems. In contrast, the open-shell theory is typical of a number of methods suggested which differ in accuracy from the viewpoint of true SCF theory, in range of applicability, complexity, and computing feasibility. A critical survey of open-shell SCF methods reported by Berthier D covers the literature up to 1962. We shall not duplicate that review here we propose rather to note some features of open-shell methods relevant to their computation feasibility and to mention procedures published after 1962. The unrestricted treatments that assume different space orbitals for different spins will be disregarded here because the restricted wave functions... 

For each k from the FEZ we calculate the elements Fpq and Spq and then solve the secular equations within the Hartree-Fock-Roothaan procedure. This step requires diagonalization" (see Appendix K, p. 982). As a result, for each k we obtain a set of coefficients c. 

Then, we calculate the elements Fp q for all atomic orbitals p,q for unit cells = 0, 1, 2,.. . y max- What is jmax The answer is certainly non-satisfactory jniax = oo- In practice, however, we often take jmax as being of the order of a few cells most often, we take max = For each from the FEZ, we calculate the elements Fpq and Spq of Eqs. (9.56) and (9.58), and then solve the secular equations within the Hartree-Fock-Roothaan procedure. This step requires diagonalization (see Appendix K available at booksite.elsevier.com/978-0-444-59436-5, p. el05). As a result, for each k we obtain a set of coefficients c for the crystal orbitals and the energy eigenvalue Sn(k). 

If the basis set used is finite and incomplete, solution of the secular equation yields approximate, rather than exact, eigenvalues. An example is the linear variation method note that (2.78) and (1.190) have the same form, except that (1.190) uses an incomplete basis set. An important application of the linear variation method is the Hartree-Fock-Roothaan secular equation (1.298) here, basis AOs centered on different nuclei are nonorthogonal. Ab initio and semiempirical SCF methods use matrix-diagonalization procedures to solve the Roothaan equations. 

Fig. 5.6 Summary of the steps in the Hartree-Fock-Roothaan-Hall SCF procedure...

Quantum mechanical calculations are carried out using the Variational theorem and the Har-tree-Fock-Roothaan equations.t - Solution of the Hartree-Fock-Roothaan equations must be carried out in an iterative fashion. This procedure has been called self-consistent field (SCF) theory, because each electron is calculated as interacting with a general field of all the other electrons. This process underestimates the electron correlation. In nature, electronic motion is correlated such that electrons avoid one another. There are perturbation procedures whereby one may carry out post-Hartree-Fock calculations to take electron correlation effects into account. " It is generally agreed that electron correlation gives more accurate results, particularly in terms of energy. 

Usually, the parent molecules M are confined to some limited size that allows rapid determination of the parent molecule density matrices within a conventional ab initio Hartree-Fock-Roothaan-Hall scheme, followed by the determination of the fragment density matrices and the assembly of the macro-molecular density matrix using the method described above. The entire iterative procedure depends linearly on the number of fragments, that is, on the size of the target macromolecule M. When compared to the conventional ab initio type methods of computer time requirements growing with the third or fourth power of the number of electrons, the linear scaling property of the ADMA method is advantageous. 

If the Fock matrix elements were infinite, then we could not manage to carry out the Hartree-Fock-Roothaan self-consistent procedure. If Et were infinite, the periodic system could not exist at all. It is, therefore, important to know when we can safely model an infinite system. 

In order to determine these unknowns the variational minimax principle of chapter 8 is invoked. For this procedure, we may again start from the energy expression of section 10.2 and differentiate it or directly insert the basis set expansion of Eq. (10.3) into the SCF Eqs. (8.185). These options are depicted in Figure 10.2. The resulting Dirac-Hartree-Fock equations in basis set representation are called Dirac-Hartree-Fock-Roothaan equations according to the work by Roothaan [511] and Hall [512] on the nonrelativistic analog. 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

Our general procedure is to represent the atoms in a molecule using the Hartree-Fock orbitals of the individual atoms occurring in the molecule. (We will also consider the interaction of molecular fragments where the Hartree-Fock orbitals of the fragments are used.) These are obtained with the above bases in the conventional way using Roothaan s RHF or ROHF procedure[45], extended where necessary. 

Rather than working with the Hartree-Fock differential equations (1.290) (with i=l,2,. ..) one usually uses the following procedure, due to Roothaan. Each orbital , is expanded in terms of some chosen complete set of one-electron basis functions gk ... 

Now, in the Hartree-Fock method (the Roothaan-Hall equations represent one implementation of the Hartree-Fock method) each electron moves in an average field due to all the other electrons (see the discussion in connection with Fig. 53, Section 5.23.2). As the c s are refined the MO wavefunctions improve and so this average field that each electron feels improves (since J and K, although not explicitly calculated (Section 5.2.3.63) improve with the i// s ). When the c s no longer change the field represented by this last set of c s is (practically) the same as that of the previous cycle, i.e. the two fields are consistent with one another, i.e. self-consistent . This Roothaan-Hall-Hartree-Fock iterative process (initial guess, first F, first-cycle c s, second F, second-cycle c s, third F, etc.) is therefore a self-consistent-field procedure or SCF procedure, like the Hartree-Fock procedure... 

Roothaan s Self-Consistent-Field Procedure.—While numerical integration techniques may be used to solve the Hartree-Fock equations in the case of atoms by the iterative method described above, the lower symmetry of the nuclear field present in molecules necessitates the use of an expansion for the determination of the molecular orbitals by a method developed by Roothaan.81 In Roothaan s approach, it is assumed that each molecular orbital may be adequately represented by a linear expansion in terms of some (simpler) set of basis functions xj, i.e. 

Ab initio MO computer programmes use the quantum-chemical Hartree-Fock self-consistent-field procedure in Roothaan s LCAO-MO formalism188 and apply Gaussian-type basis functions instead of Slater-type atomic functions. To correct for the deficiencies of Gaussian functions, which are, for s-electrons, curved at the nucleus and fall off too fast with exp( —ar2), at least three different Gaussian functions are needed to approximate one atomic Slater s-function, which has a cusp at the nucleus and falls off with exp(— r). But the evaluation of two-electron repulsion integrals between atomic functions located at one to four different centres is mathematically much simpler for Gaussian functions than for Slater functions. 

There are both numerical and analytical ways of carrying out this procedure. The first is the easiest to understand and was first applied to the Hartree-Fock (HF) method by Cohen and Roothaan[23]. One simply takes various values of F (usually of the order of 0.001 au), finds the corresponding EF and makes a fit to Eq. (9). This is called the finite field method and it may be applied within the framework of any of the standard methods which determine energies, e.g., HF, MP2, MP4, coupled cluster (CC), MCSCF. 

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