By Ernest Roy Davidson
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Additional info for Reduced Density Matrices in Quantum Chemistry
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It is fairly easy to carry out for p as well as p,. Also, the general result (4-54) is well adapted for carrying out perturbation theory with a spin dependent opera tor. It will be noticed that in order to treat such perturbations, only one component (with Μ φ 0) of each degenerate set of states Ψ is needed in order to evaluate all matrix elements. From (4-41) it follows that if Ψ is an eigenfunction of S , so are the natural spin orbitals | γ / ) . Further, unless M = 0, the natural spin orbitals of a spin and β spin are different and are M dependent.
35, 668 (1963). Davidson, E. , In "Physical Chemistry," Vol. Ill, Electronic Structure of Atoms and Molecules (D. ), p. 113. Academic Press, New York, 1969. Garrod, C , and Perçus, J. , / . Math. Phys. 5, 1756 (1964). , J. Math. Phys. 8, 2063 (1967). " Academic Press, New York, 1970. Smith, D. , In "Reduced Density Matrices with Applications to Physical and Chemical Systems" (A. J. Coleman and R. M. ), Queen's Papers Pure Appl. Math. 11, 169 (1968). Physical Properties of the First-Order Reduced Density Matrix It has long been assumed that the charge and bond order matrix (first-order reduced density matrix in modern terminology) contains most of the essential information about chemical bonding and electron distribu tion in a molecule.
That is, the sum of the squares of the coefficients of all Slater determinants involving a particular natural orbital is as large as possible (subject to the condition that the ith natural orbital is orthogonal to all preceding ones). Also, the total contribution from all natural orbitals beyond the zth one is as small as possible. It has already been shown that if the rank r is N, then p, is a Fock-Dirac density and Ψ is a single Slater determinant. Now let us consider the case r = Ν + 1. Without loss of generality, let us assume that the Slater determinants are already based on natural orbitals (if not, the natural orbitals can be found and the wavefunction rewritten using them).