Coulombic energy exchange

Number of cl Electrons Weak Field Arrangement LFSE (X) Coulombic Energy Exchange Energy ... 

Thus E. is the average value of the kinetic energy plus the Coulombic attraction to the nuclei for an electron in ( ). plus the sum over all of the spin orbitals occupied in of the Coulomb minus exchange interactions. If is an occupied spin orbital, the temi [J.. - K..] disappears and the latter sum represents the Coulomb minus exchange interaction of ( ). with all of the 1 other occupied spin orbitals. If is a virtual spin orbital, this cancellation does not occur, and one obtains the Coulomb minus exchange interaction of cji. with all N of the occupied spin orbitals. 

These two expressions differ in a very important way the sum of occupied orbital energies double counts the Coulomb minus exchange interaction energies. Thus, within the FIE approximation, the sum of the occupied orbital energies is not equal to the total energy. 

A I lai lcee-Fock calculation provides a set of orbital energies, e,. What is the significance oi these The energy of an electron in a spin orbital is calculated by adding the core inleraclion to the Coulomb and exchange interactions with the other electrons in the svstein ... 

The HF equations F ( )i = 8i (jii comprise a set of integro-differential equations their differential nature arises from the kinetic energy operator in h, and the coulomb and exchange operators provide their integral nature. The solutions of these equations must be... 

The operator hi is a one-electron operator, representing the kinetic energy of an electron and the nuclear attraction. The operators J and K are called the Coulomb and exchange operators. They can be defined through their expectation values as follows. 

For the purpose of deriving the variation of the energy, it is convenient to express the energy in terms of Coulomb and Exchange operators. 

Within the hydration process in Eqn. (8.8), a spherical ion becomes a (hydrated) octahedral ion, [M(H20)6]. Part of the Coulomb energy of the free ion concerns repulsion and exchange terms within the d" configuration. This is replaced by equivalent repulsion and exchange terms within the configuration. Let us estimate the trends in these quantities separately. 

Here ho is the kinetic energy and nuclear attraction operator while and 1C are the coulomb and exchange operators, respectively. The coefficients X and Y are solutions of the RPA equations, which for the / singlet transition with excitation energy can be written as... 

Table 13-8. Self-interaction error components for Coulomb and exchange energies (Ej + Ex) as well as for the correlation energy (Ec), and the resulting sum for the H atom, the H2 molecule, and the H3 transition structure [kcal/mol]. Data taken from Csonka and Johnson, 1998.

In earlier sections, we have seen that the energy of a molecular orbital can be expressed in terms of the coefficients, a and the Coulomb and exchange integrals. For the second wave function, this can be expressed as... 

Using an anti-symmetrized, rather than a simple product of one-electron orbitals, the expectation value of the energy becomes a sum of one-electron, Coulomb and exchange terms ... 

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