Orbital expansion coefficients

The problem has now become how to solve for the set of molecular orbital expansion coefficients, c. . Hartree-Fock theory takes advantage of the variational principle, which says that for the ground state of any antisymmetric normalized function of the electronic coordinates, which we will denote H, then the expectation value for the energy corresponding to E will always be greater than the energy for the exact wave function ... 

The variational principle leads to the following equations describing the molecular orbital expansion coefficients, c. , derived by Roothaan and by Hall ... 

Both the Fock matrix—through the density matrix—and the orbitals depend on the molecular orbital expansion coefficients. Thus, Equation 31 is not linear and must be solved iteratively. The procedure which does so is called the Self-Consistent Field... 

So far, we have considered only the restricted Hartree-Fock method. For open shell systems, an unrestricted method, capable of treating unpaired electrons, is needed. For this case, the alpha and beta electrons are in different orbitals, resulting in two sets of molecular orbital expansion coefficients ... 

From molecular orbital theory, a many-electron wavefunction, may be defined by a determinant of molecular wavefunctions, i/ ,. The i[i, may in turn be expressed as a linear combination of one-electron functions, that is, (fj, = where cM, are the molecular orbital expansion coefficients... 

It is not difficult to see that the problem of simultcmeous optimization of linear and non-linear parameters addressed by us eamller in this section has a direct bearing on the MC-SCF method. The only difference lies in the fact that in the conventional MC-SCF scheme, one expands the MC-SCF orbitals in terms of a finite basis set and optimizes the orbital expansion coefficients and not the exponents, to get at the optimal orbital... 

In the MCSCF method many of the operations required in the formal development and in the actual computational implementation involve the use of linear algebra. These manipulations of matrices and vectors are discussed in this section and some necessary background for later discussions is introduced. The first reason that the manipulation of matrices is important in the MCSCF method is that the molecular orbitals (MOs) used to define the wavefunction are expanded in an atomic orbital (AO) basis. The orbital expansion coefficients may be collected into the matrix C and the relation between the two orbital sets may be written as... 

The set of variables in the MCSCF optimization process determine the changes in the CSF mixing coefficients and the orbital expansion coefficients during each iteration. When particular choices of CSFs are employed in the wavefunction expansion or when particular relations between the CSF expansion coefficients are satisfied, redundant variables will occur within this... 

Because of the dependence of the determinants forming this set on the orbital expansion coefficients as well as on the orbital exponents, it is clear that the optimal determinant which minimizes the kinetic energy for the noninteracting system can be reached by varying these orbital parameters. At the extremum of this constrained variation we have, therefore, the following inequality for the kinetic energy functional ... 

The i are the orbital energies and the c i are the molecular orbital expansion coefficients in terms of the atomic orbital basis set Xp- Contributions from terms greater than second order are generally small and can for most chemical applications be neglected [41, 42]. Only a limited number of applications of P(r) for analysis of intermolecular interactions have appeared in the literature. The most common approach of analysis has been to calculate a total interaction index, a "polarization-corrected electrostatic potential", defined by... 

In this paper, the systematic truncation of a distributed universal even-tempered basis set capable of supporting high precision in both mauix Haitree-Fock and second order many-body perturbation theory calculations is explored using the ground state of the boron fluoride molecule as a prototype. The truncation procedure adopted is based on the magnitude of the orbital expansion coefficients associated with a given basis function in each of the occupied orbitals. 

The truncation procedure explored in the present smdy is described in detail in section 2. An analysis of the orbital expansion coefficients for the ground state of the BF molecule is presented in section 3, where the truncated basis sets employed in the present study are defined. The results of both matrix Hartree-Fock calculations and second-order many-body perturbation theory studies are given in section 4 together with a discussion of the properties of the truncated basis sets. The final section, section 5, contains a discussion of the results and conclusions are given. 

In this work, n is taken to be an integer, but, in general, fractional values can be considered. For a given t each of the basis subsets, Sx /) can be divided into two sets for each occupied orbital, p one for which the magnitude of the orbital expansion coefficients, Cpp, is equal to or exceeds r ... 

A detailed analysis of the orbital expansion coefficients for the basis subsets of p-type and higher symmetries centred on the B atom is presented in Table 2. The original subsets, Sb(p), SB(d) and 5b(/), each contained 15 primitive Gaussian functions. The truncation of these subsets for Tg, T5, T4 and T3 is shown in Table 2. For Tg subsets Sb p), SB d) and Sb(/) contained 13, 11 and 10 primitive Gaussian functions, respectively, whilst for there are 13, 9 and 8 functions and for T4 there are 10, 7 and 6 functions. For the most heavily truncated set considered, with T3, there are 8, 6 and 3 functions, respectively, in the subsets Sb(p), SB(d) and SB(f). In all of the truncated subsets the number of functions decreases with increasing symmetry type, /. 

In Table 3, a detailed analysis of the orbital expansion coefficients is reported for the subset containing functions of x-type centred on the F nucleus. Again the functions retained in the truncated subsets are indicated by the dots. The numbers of functions in each of the subsets for Too, 6, T5, T4 and T3 are 30, 26, 24, 20 and 17. In each of the truncated sets there are more functions surviving the condition (5) than in the corresponding sets centred on the B nucleus. This reflects the higher nuclear charge in F which necessitates the retention of more contracted primitive functions. 

We present an analysis of the orbital expansion coefficients related to the functions of p- and higher symmetry centred on the bond mid-point in Table 6. The most diffuse functions from the origin subsets containing 15 functions of each symmetry type were... 

The coelTicient matrix C whose elements are the orbital expansion coefficients then becomes... 

Here are the molecular orbital expansion coefficients for the basis functions < ). The problem of determining the molecular orbital has been reduced from finding a complete description of the three-dimensional function i , to finding only a finite set of linear coefficients for each orbital. If the basis functions 4) are atomic orbital functions, then this linear expansion is known as a linear combination of atomic orbitals (LCAO). 

We now add contributions corresponding to treating the orbital expansion coefficients as variables with fictive masses Pi. 

The Fock matrix represents the average effects of the field of all the electrons and nuclei in each orbital as the orbitals depend on the molecular orbital expansion coefficients, thus the self-consistent method (SCF) is used to solve... 

In this equation, the coefficients c j are called the molecular orbital expansion coefficients, and are optimized during the computational procedme. In chemical terms, one can think of the basis functions as the sets of constituent atomic orbitals, which mix to form the molecular orbitals of the molecule. In order to approach the exact solution to the Schrodinger equation, an infinite set of basis functions would be required, as this would introduce sufficient mathematical... 

The coefficients in this expression d p) are fixed for the basis set of a given atom, and should not be confused with the molecular orbital expansion coefficients iCpi) of equation 6, which are determined for a given system during the ab initio... 

Differentiation of the basis set form of SCF, the Hartree-Fock-Roothaan equation, is complicated by the fact that the Fock operator is itself dependent on the orbital set. But this complication is not difficult to deal with. We will use C to represent the matrix of orbital expansion coefficients, S to be the matrix (operator) of the overlap of basis functions, F to be the Fock operator matrix, and E to be the orbital eigenvalue (orbital energy) matrix. The equation to be differentiated is... 

The procedure for determining the electron density and the energy of the system within the DFT method is similar to the approach used in the Hartree-Fock technique. The wavefunction is expressed as an antisymmetric determinant of occupied spin orbitals which are themselves expanded as a set of basis functions. The orbital expansion coefficients are the set of variable parameters with respect to which the DFT energy expression of equation 15 is optimized. The optimization procedure gives rise to the single particle Kohn-Sham equations which are similar, in many respects, to the Roothaan-Hall equations of Hartree-Fock theory. 

Implementation of the algebraic approximation requires the determination of the orbital expansion coefficients, in eq. (3.140). This is achieved by solving the matrix Hartree-Fock equations which may be written... 

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