Closed-shell molecule, self-consistent

The solution of the secular equation Fy —F5y = 0 requires the evaluation of the constituent matrix terms Fy. The Fy s are, however, themselves functions of the coefficients of the atomic orbitals amt through Pjel and therefore can only be evaluated by solving the secular equation. The Hartree-Fock procedure thus requires that a preliminary guess be made as to the values of the molecular population distribution terms Pici these values are then used to calculate the matrix elements Fy and thence solve the secular determinant. This, in turn, provides a better approximation to the wave function and an. .improved set of values of Pm. The above procedure is repeated with this first improved set and a second improved set evaluated. The process is repeated until no difference is found between successive improved wave functions. Finally, it may be shown that when such a calculation has been iterated to self-consistency the total electronic energy E of a closed shell molecule is given by... 

G. Malli and J. Oreg, ]. Chem. Phys., 63, 830 (1975). Relativistic and Self-Consistent Field Theory for Closed-Shell Molecules. 

From Figures 8.1(a) and 8.1(b) it would appear that the energy required to eject an electron from an orbital (atomic or molecular) is a direct measure of the orbital energy. This is approximately true and was originally proposed by Koopmans whose theorem can be stated as follows Tor a closed-shell molecule the ionization energy of an electron in a particular orbital is approximately equal to the negative of the orbital energy calculated by a self-consistent field (SCF see Section 7.1.1) method , or, for orbital i,... 

There is generally little ambiguity associated with calculations of molecular orbitals for closed-shell molecules. The Hartree-Fock method (as appoximated by self-consistent field calculations in necessarily finite atomic orbital basis sets11) provides a solution that obeys some of the symmetry properties that must... 

Mohanty, A. and Clementi, E. (1990) Dirac-Fock self-consistent field calculations for closed-shell molecules with kinetic balance and finite nuclear size. In Modem Techniques in Computational Chemistry MOTECC-90 (ed. E. Clementi), pp. 693—730. ESCOM, Leiden. Mohanty, A. K. and Clementi, E. (1991) Int. J. Quant. Chem 39,487-517. 

Pisani, L. and Clementi, E. (1993) Dirac-Fock self-consistent field calculations for closed-shell molecules with kinetic balance. In Gementi (1993), pp. 345-360. 

A. Mohanty, E. dementi, Dirac-Fock Self-Consistent Field Calculations for Closed-Shell Molecules with Kinetic Balance and Finite Nuclear Size, in E. dementi (Ed.), Modern Techniques in Computational Chemistry MOTECC-90, ESCOM, Leiden, 1990, pp. 693-730. 

In general, a qualitatively correct description of the ground state of a closed-shell molecule is provided by a single Slater determinant. This is why semiempirical (one-determinant) self-consistent field (SCF) methods can be applied quite successfidly to the determination of ground-state properties such as geometries, vibrational frequencies, and relative energies. Many electronically excited states, however, contain more then one dominant configuration state function. The simplest description of an excited state is the orbital picture where one electron has been moved from an occupied to an... 

We decided to undertake this with the aim of formulating a generally applicable computational scheme for radicals which would be a natural extension of the PPP and semiempirical all-valence-electron methods for closed-shell molecules. We used for this purpose the self-consistent-field open-shell methods of Longuet-Higgins and Pople [2] and of Roothaan [3], we derived all expressions necessary for CI-S calculations [4, 5], and we tested the semiempirical open-shell PPP-type and INDO/S calculations systematically for various classes of radicals. 

The emergence of complete systems for self-consistent-field (SCF) calculations on small and medium-sized closed-shell molecules. 

When Pji in Eq. [2] represents the ground state (where, in a closed-shell molecule, each of the N/2 lowest energy MOs is occupied by two electrons), we denote it by the symbol Pq. If a set of basis functions, (j) is chosen such that it allows sufficient flexibility in the set of MOs, j/, a single Slater determinant Pq may indeed serve as a good basis to describe different properties of closed-shell molecules. The optimal AO coefficients in the MOs / that are occupied in Pq can be found by carrying out a HE self-consistent field (SCF) calculation. ... 

The next step is to make the Hartree-Fock self-consistent field (HF-SCF) approximation as described previously for a multi-electron atom in Section 8.4. The Hartree-Fock approximation results in separation of the electron motions resulting (along with the Pauli principle) in the ordering of the electrons into the molecular orbitals as shown in Figure 9-5 for carbon monoxide. Hence, the many-electron wavefunction i for an N-electron molecule is written in terms of one-electron space wavefunctions,/, and spin functions, a or p, like what was done for complex atoms in Section 8.4. At this stage it is assumed that the N-electron molecule is a closed-shell molecule (all the electrons are paired in the occupied molecular orbitals). How molecules with open shells are represented will be discussed later in this Section. 

In the later part of the 1950 s, It was evident that it was necessary to distlngush the new approach dealing with different orbitals for a-spln and P-spln from the previous approach starting out from symmetry restrictions the latter was called the Restricted Hartree-Fock (RHF) scheme, whereas the new approach was called the Unrestricted Hartree-Fock (UHF) scheme. For some time there was a certain amount of competition between the two schemes. In the late 1950 s, it was further shown that the RHF-scheme for closed-shell systems was completely se[f-consistent not only for atoms but also for molecules and solids [16.17] and that, if one started by imposing a symmetry requirement on the original Slater determinant, this assumption would be self-consistent, i.e. the final determinant would have the same symmetry property. Since symmetry properties are of such fundamental importance in quantum theory, one would hence anticipate that the RHF-scheme would... 

This factorization amounts to the statement thatEq. (2.25) breaks down into two separate linear systems, one for the determination of a orbitals, and the other for n orbitals. In the Hartree-Fock scheme, self-consistent field equations (SCF equations 2.25) have as solutions symmetry-adapted functions (i.e. in the case of planar unsaturated molecules symmetric or antisymmetric functions with respect ot the molecular plane), at least for closed-shell ground states iM8,20,2i) ... 

A key development in quantum chemistry has been the computation of accurate self-consistent-field wave functions for many diatomic and polyatomic molecules. The principles of molecular SCF calculations are essentially the same as for atomic SCF calculations (Section 11.1). We shall restrict ourselves to closed-shell configurations. For open shells, the formulas are more complicated. 

Kim has formulated a relativistic Hartree-Fock-Roothaan equation for the ground states of closed-shell atoms using Slater-type orbitals. Relativistic effects in atoms have been reviewed by Grant. Malli and coworkers have formulated a relativistic SCF method for molecules. In this method, four-component spinor wavefunctions are obtained variationally in a self-consistent scheme using Gaussian basis sets. 

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