Analytical Chemistry of Liquid Fuel Sources by Uden P.C., Siggia S., Jensen H.B. (eds.)

By Uden P.C., Siggia S., Jensen H.B. (eds.)

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We recall that forming a Slater determinantal wavefunction with orbitals that are obtained by a linear transformation yields just the Slater determinant of the original orbitals multiplied by the determinant of the transformation matrix: IyI = lcpl det T. ) So the Slater determinantal wavefunction for triplet H2 on the basis of the orthogonal set (yig,yiI1)is just the antisymmetrized and renormalized product wavefunction Yo = NA{qAqB}: l/m Yo = lyig~wuctl= IcpAaq,aIdetT = NA{cpAacp,a} [I61 Note that the additional normalization factor det T appears because the qAand cpB in the Slater determinant are not orthogonal; therefore, Iq,acp,al is not is not equal normalized.

Asterisks indicate the positions of the nuclei. viewed as a manifestation of the Pauli principle. According to this principle, which follows from the antisymmetry requirement for fermion wavefunctions, electrons are not allowed to be at the same place with the same spin. The antisymmetrization we had to carry out actually reduces the probability density in the overlap regionso from what it would be if the necessary antisymmetry of the wavefunction had not been taken into account. 18 Kohn-Sham Density Functional Theory The energy change from E A + E B to EO can be written as the electrostatic interaction defined above, plus all other effects lumped together into the Pauli repulsion term AEPauli: It is always possible to write the energy corresponding to a wavefunction as the sum of the kinetic energy (the expectation value of the kinetic energy operator) and of the potential energy, which is the expectation value of all Coulombic operators, so we write AEO as the sum of kinetic and potential energy differences, AEO = AVO + AT0 = AVelstat + AVpauli + A P ~ 4 1 AEPauliis broken up into a potential energy (AVpauli)and a kinetic energy ( A P ) part.

Viewed as a manifestation of the Pauli principle. According to this principle, which follows from the antisymmetry requirement for fermion wavefunctions, electrons are not allowed to be at the same place with the same spin. The antisymmetrization we had to carry out actually reduces the probability density in the overlap regionso from what it would be if the necessary antisymmetry of the wavefunction had not been taken into account. 18 Kohn-Sham Density Functional Theory The energy change from E A + E B to EO can be written as the electrostatic interaction defined above, plus all other effects lumped together into the Pauli repulsion term AEPauli: It is always possible to write the energy corresponding to a wavefunction as the sum of the kinetic energy (the expectation value of the kinetic energy operator) and of the potential energy, which is the expectation value of all Coulombic operators, so we write AEO as the sum of kinetic and potential energy differences, AEO = AVO + AT0 = AVelstat + AVpauli + A P ~ 4 1 AEPauliis broken up into a potential energy (AVpauli)and a kinetic energy ( A P ) part.

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