Self-consistent field cycles

Elapsed time (sec) for the full self-consistent field cycle. 

To construct the Fock matrix, one must already know the molecular orbitals ( ) since the electron repulsion integrals require them. For this reason, the Fock equation (A.47) must be solved iteratively. One makes an initial guess at the molecular orbitals and uses this guess to construct an approximate Fock matrix. Solution of the Fock equations will produce a set of MOs from which a better Fock matrix can be constructed. After repeating this operation a number of times, if everything goes well, a point will be reached where the MOs obtained from solution of the Fock equations are the same as were obtained from the previous cycle and used to make up the Fock matrix. When this point is reached, one is said to have reached self-consistency or to have reached a self-consistent field (SCF). In practice, solution of the Fock equations proceeds as follows. First transform the basis set / into an orthonormal set 2 by means of a unitary transformation (a rotation in n dimensions),... 

The process is continued for k cycles till we have a wavefunction and/or an energy calculated from that are essentially the same (according to some reasonable criterion) as the wavefunction and/or energy from the previous cycle. This happens when the functions i/ (l), i//(2),. .., j/(n) are changing so little from one cycle to the next that the smeared-out electrostatic field used for the electron-electron potential has (essentially) ceased to change. At this stage the field of cycle k is essentially the same as that of cycle k — 1, i.e. it is consistent with this previous field, and so the Hartree procedure is called the self-consistent-field-procedure, which is usually abbreviated as the SCF procedure. 

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... 

The problem is now solved again by an iterative process, which starts with a choice of the x set and the expansion (6.58). The Hartree-Fock operator F and the matrix representation Fx are calculated, (6.64) is solved for the orbital energies, and these are used to compute a new set of coefficients in (6.63). If these are different from the starting set, the cycle is repeated until the self-consistent-field limit is reached. The total electronic energy is obtained by adding the SCF energy to the core energy for the lowest occupied n/2 levels ... 

The next step is to do the same type of calculation to obtain a new wave function for electron 2 moving in a field of electrons described by the wave functions ip i, ip3, tfa,. .., ipN. This step leads to a new function ip2 for electron 2. Now the process is carried out for electron 3 interacting with electrons described by the wave functions ip i, ip2, wave functions ip2, ip2, ip 3,..., ip N. Then the entire process starts again with electron 1 and continues through electron N to give the new functions ipY, ip2, ip3,..., ipu- The procedure is diagrammed in Fig. 12.32. When a given cycle produces a set of wave functions that are virtually identical to the previous set, a self-consistent field is achieved and the procedure is terminated. 

Appropriate geometries of both HCl and HF molecules were fixed by calculation with the Eigenvector Following method in MOPAC with various Hamiltonians of AMI [11], PM3 [12], and PMS. The optimization of the state of each molecule was started at the point of initially defaulted value of inter-atomic distance. The calculation was carried out until the cutoff value of less than 1.000 in gradient by root-mean-square (RMS) where the value less than 1.000 means to achieve the self-consistent field (SCF). Tentative heat of formation, AH was obtained by MOPAC calculation. Results are listed in Table 1. In the case of HCl, the cutoff value by AMI reached to the value of less than 1.000 in gradient only in 3 cycles of optimization, and the value of AH was -24.61233 kcal moF with the value of 1.2842 A of inter-atomic distance. Values of AH were obtained as -20.46808 and -30.41903 kcal moF by PM3 and PMS, respectively. In the case of HF, the value by AMI reached to -74.28070 kcal moF in 6 cycles of optimization with the value of 0.8265 A of inter-atomic distance. AH values were obtained as -62.75007 and -67.15007 kcal moF by PM3 and PMS, respectively. Geometries of both HCl and HF by three Hamiltonians were detennined by these optimizations. 

In general the final potential-energy functions are not identical with those chosen initially. The cycle is then repeated, using the results of step 4 as an aid in the estimation of new potential-energy functions. Ultimately a cycle may be carried through for which the final potential-energy functions are identical (to within the desired accuracy) with the initial ones. The field corresponding to this cycle is called a self-consistent field for the atom. 

Hartree and Fock (Fock 1930 Hartree 1928) formalism uses the single SD form of the total wave function, which is solved under self-consistent field (SCF) approximation. This involves an iterative process in which the orbitals are improved cycle to cycle until the electronic energy reaches a constant minimum and the orbitals no longer change. Upon convergence of the SCF method, the minimum-energy MOs produce an electric field that generates the same orbitals and hence the self-consistency. 

The numerical integration also can be used to calculate the matrix elements of the exchange-correlation potential. For the numerical integration, the atomic partition method proposed by Savin [392] and Becke [393] has been adopted and combined with Gauss-Legendre (radial) and Lebedev (angular) quadratures [394]. The Kohn-Sham LCAO periodic method based on numerical integration at each cycle of the self-consistent-field process is computationally more expensive than the periodic LCAO Hartree-Fock method that is almost fully analytical. 

Both types of procedure typically adopt a self-consistent field (SCF) procedure, in which an initial guess about the composition of the LCAO is successively refined until the solution remains unchanged in a cycle of calculation. For example, the potential energy of an electron at a point in the molecule depends on the locations of the nuclei and all the other electrons. Initially, we do not know the locations of those electrons (more specifically, we do not know the detailed form of the wavefunctions that describe their locations, the molecular orbitals they occupy). First, then, we guess the form of those wavefunctions—we guess... 

In the self-consistent field procedure, an initial guess about the composition of the molecular orbitals is successively refined until the solution remains unchanged in a cycle of calculations. 

The occupied molecular orbitals allow us to define the Pq density matrix as in any SCF computational scheme. The total density matrix P (equation 28) is then defined and the Fockian (equation 29) can be computed for another cycle of the local self-consistent field (LSCF) method. One notices that this scheme exhibits little difference with the usual SCF computations. In particular, the B matrix plays the same role as the orthogonalization matrix in the usual methods. This computational scheme is therefore rather easy to implement. 

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