Excited states and solvatochromic shifts within a nonequilibrium solvation approach: A new formulation of the integral equation formalism method at the self-consistent field, configuration interaction, and multiconfiguration self-consistent field level

1998 ◽  
Vol 109 (7) ◽  
pp. 2798-2807 ◽  
Author(s):  
Benedetta Mennucci ◽  
Roberto Cammi ◽  
Jacopo Tomasi
Keyword(s):  
Integral Equation ◽  
Excited States ◽  
Field Configuration ◽  
The Self ◽  
Consistent Field ◽  
Self Consistent ◽  
Solvatochromic Shifts ◽  
Self Consistent Field ◽  
Nonequilibrium Solvation ◽  
New Formulation

scholarly journals Self-consistent field, with exchange, for beryllium - II—The (2 s ) (2 p ) 3 P and 1 P excited states

1936 ◽  
Vol 154 (883) ◽  
pp. 588-607 ◽  
Keyword(s):  
Excited States ◽  
Normal State ◽  
Wave Functions ◽  
The Self ◽  
Energy Separation ◽  
Radial Wave ◽  
Consistent Field ◽  
Exchange Effects ◽  
Self Consistent ◽  
Self Consistent Field

In a recent paper we gave an account of the method and results of the solution of Fock’s equations of the self-consistent field, including exchange effects, for the normal state of neutral beryllium. The present paper is concerned with the extension of the calculations to the first two excited states, (1 s ) 2 (2 s ) (2 p ) 3 P and 1 P, of the same atom. This extension was undertaken for two reasons. Firstly, before going on to attempt the solution of Fock’s equations for a heavier atom, we wished to get some experience of the process of solution of Fock’s equations for a configuration involving wave functions which overlap to a greater extent than the wave functions (1 s ) and (2 s ) of the normal state, and for which exchange effects might be expected to be corre­spondingly greater; and secondly, for an atom with more than one electron outside closed ( nl ) groups, so that a given configuration gives rise to more than one term, the equations of the self-consistent field, when exchange effects are included, are no longer the same for the different terms, and it seemed likely to be of interest to examine the consequent difference between the radial wave functions for the different terms (here 3 P and 1 P), and the effect of this difference on the calculated energy separation between the terms.

scholarly journals Self-consistent field, with exchange, for Cl -

1936 ◽  
Vol 156 (887) ◽  
pp. 45-62 ◽  
Keyword(s):  
Excited States ◽  
The Self ◽  
The Other ◽  
Negative Ion ◽  
Field Problem ◽  
Consistent Field ◽  
Exchange Effects ◽  
Self Consistent ◽  
Self Consistent Field ◽  
The Difference

Quantitative solutions by Fock's equations for the self-consistent field, including exchange effects, have now been obtained for the normal state of Na + , and for the normal and some excited states of neutral Li and Na, by Fock and Petrashen, and for the normal and the (2 s ) (2 p ) 3 P and 1 P excited states of neutral Be, by the present authors. The work here described was undertaken as the first step in carrying out the solution of Fock's equations for heavier atoms; calculations for Cu + , in progress at the time of writing, will carry this a step further. The effect of the inclusion of exchange terms on the self-consistent field may be expected to be particularly large for a negative ion, on account of the sensiticeness of the wave function of the outer ( nl ) group of such an ion (and, to a less extent, of those of the other groups of the outer shell also). This, of course, is likely to make the process of solution of Fock's equations more than usually difficult and lengthy for a negative ion, and made the project of obtaining such a solution appear somewhat ambitions; but, on the other calculations, are greatest when the difference from the solution of the self-consistent field problem without exchange is greatest, and for this reason it seemed desirable to carry out the solution of Fock's equations for at least one negative ion.

scholarly journals The relativistic self-consistent field

1935 ◽  
Vol 152 (877) ◽  
pp. 625-649 ◽  
Keyword(s):  
Wave Equation ◽  
Field Equation ◽  
Wave Functions ◽  
The Self ◽  
Magnetic Interactions ◽  
Consistent Field ◽  
Exchange Effects ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Schrödinger's Equation

1—The method of the self-consistent field for determining the wave functions and energy levels of an atom with many electrons was developed by Hartree, and later derived from a variation principle and modified to take account of exchange and of Pauli’s exclusion principle by Slater* and Fock. No attempt was made to consider relativity effects, and the use of “ spin ” wave functions was purely formal. Since, in the solution of Dirac’s equation for a hydrogen-like atom of nuclear charge Z, the difference of the radial wave functions from the solutions of Schrodinger’s equation depends on the ratio Z/137, it appears that for heavy atoms the relativity correction will be of importance; in fact, it may in some cases be of more importance as a modification of Hartree’s original self-nsistent field equation than “ exchange ” effects. The relativistic self-consistent field equation neglecting “ exchange ” terms can be formed from Dirac’s equation by a method completely analogous to Hartree’s original derivation of the non-relativistic self-consistent field equation from Schrodinger’s equation. Here we are concerned with including both relativity and “ exchange ” effects and we show how Slater’s varia-tional method may be extended for this purpose. A difficulty arises in considering the relativistic theory of any problem concerning more than one electron since the correct wave equation for such a system is not known. Formulae have been given for the inter-action energy of two electrons, taking account of magnetic interactions and retardation, by Gaunt, Breit, and others. Since, however, none of these is to be regarded as exact, in the present paper the crude electrostatic expression for the potential energy will be used. The neglect of the magnetic interactions is not likely to lead to any great error for an atom consisting mainly of closed groups, since the magnetic field of a closed group vanishes. Also, since the self-consistent field type of approximation is concerned with the interaction of average distributions of electrons in one-electron wave functions, it seems probable that retardation does not play an important part. These effects are in any case likely to be of less importance than the improvement in the grouping of the wave functions which arises from using a wave equation which involves the spins implicitly.

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