Orbital representation, diagonal elements

G. Constraints on Off-Diagonal Elements from Other Positive-Definite Hamiltonians Linear Inequalities from the Spatial Representation Linking the Orbital and Spatial Representations... 

For simplicity, we shall commonly refer to the Q-electron distribution function as the 2-density and the 2-electron reduced density matrix as the 2-ntatrix. In position-space discussions, the diagonal elements of the 2-ntatrix are commonly referred to as the 2-density. In this chapter, we will also refer to the diagonal element of orbital-space representation of the Q-vaatnx as the 2-density. 

Note that there is no simple method for converting the g-density from the spatial representation to the orbital representation and back. The g-density in the spatial representation depends on off-diagonal elements of the g-matrix in the orbital representation and the g-density in the orbital representation depends on off-diagonal elements of the g-matrix in the spatial representation. 

The (Q, R) conditions form a polyhedral hull containing the set of A-representable rg. The (Q, R) conditions either contain, or imply, every necessary condition that can be stated for the diagonal elements of the g-matrix without using more than R distinct orbital indices. In this sense, the (Q, R) conditions are the complete set of R-orbital necessary conditions for the g-density. 

Tr PrJ Pr > 0 represents the complete set of constraints on the diagonal elements of the l -matrix that can be expressed using no more than R distinct orbital indices. The (R, R) conditions are necessary, but not sufficient, for the A-representability of the l -matrLx. 

Recall that the (2, 2) conditions contain all the A -representability constraints on the diagonal element of the 2-matrix that can be stated without reference to more than two orbitals. 

The orbital representation is not used in most of the recent work on computational methods based on diagonal elements of density matrices. This is partly for historical reasons—most of the work has been done by people trained in density functional theory—and partly this is because most of the available kinetic energy functionals are known only in first-quantized form. For example, the popular generalized Weisacker functional [2, 7-11],... 

Recall that the spatial representation of the g-density actually depends on the off-diagonal elements of the density matrix in the orbital representation. (See Eqs. (18)-(21) and the surrounding discussion.) This suggests that some progress can be made by using the A -representabihty constraints for off-diagonal elements in the density matrix. If one chooses the one-electron Hamiltonian associated with the G condition to be a simple function, then one finds that [22, 28, 54]... 

In a basis set that contains K spin orbitals, the Q, K) conditions are necessary and sufficient for the A-representability of the diagonal elements of the 2-matrix, but only necessary for the off-diagonal elements, r,j. ... 

From Table 7-9,2 and using eqn (5-7,2) we can find the diagonal elements of the matrices which represent the 4h point group in the p-orbital basis and in the d-orbital basis. From these elements we get the characters of two reducible representations they are shown in Table 7-9,3, By applying eqn (7-4.2)... 

When compared with the diagonal element of the spin orbit operator in the effective Hamiltonian, AAE in a Hund s case (a) representation, we see that... 

In low S3mimetries such as D2ft, Ds, and C2 where two (or more) sets of f-orbitals belong to the same irreducible representation it is necessary to solve a secular determinant. The off-diagonal elements will be given by ... 

The reduced density matrix can be converted into a discrete representation that involves sums over all the Slater determinants, MOs, and basis functions. This matrix will in general have many off-diagonal elements. The matrix is Hermi-tian therefore it can be diagonalized. The orbitals that result from the diagonal reduced-density matrix are called natural orbitals, and the diagonal elements are the occupation numbers for these orbitals. The natural orbitals are orthonormal molecular orbitals having maximal occupancy. 

The diagonal elements of this matrix give the probability density for the electrons in a point r with the spin s. A matrix representation of the density matrix can be obtained by an expansion in the basis of SOs used to construct the total wave function (orbitals which... 

In fact, as is familiar from the use of point groups to generate symmetry orbitals (Chapter 18), it is only necessary to know the diagonal elements of the representation matrices to form a projection operator which will generate a suitable function, in our case the projector... 

Multiplying (3.70) by matrix representation of the Fock operator in the basis of spin orbital eigenfunctions is diagonal with diagonal elements equal to the orbital energies. 

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