Diagonal matrices, expectation values

The quantum-mechanical state is represented in abstract Hilbert space on the basis of eigenfunctions of the position operator, by F(q, t). If the eigenvectors of an abstract quantum-mechanical operator are used as a basis, the operator itself is represented by a diagonal square matrix. In wave-mechanical formalism the position and momentum matrices reduce to multiplication by qi and (h/2ni)(d/dqi) respectively. The corresponding expectation values are... 

The preceding is a rather comprehensive—but not exhaustive— review of N-representability constraints for diagonal elements of reduced density matrices. The most general and most powerful V-representability conditions seem to take the form of linear inequalities, wherein one states that the expectation value of some positive semidefinite linear Hermitian operator is greater than or equal to zero, Tr [PnTn] > 0. If Pn depends only on 2-body operators, then it can be reduced into a g-electron reduced operator, Pq, and Tr[Pg vrg] > 0 provides a constraint for the V-representability of the g-electron reduced density matrix, or 2-matrix. Requiring that Tr[Pg Arrg] > 0 for every 2-body positive semidefinite linear operator is necessary and sufficient for the V-representability of the 2-matrix [22]. 

Note that we introduced the superscript CD in order to distinguish the expressions obtained by Clark and Davidson from those by Mayer, which will be given in the following marked by Ma. In a similar fashion, Mayer s partitioning of the total spin expectation value can be derived. Starting from Lowdin s expression for the total spin expectation value, Eq. (96), a one-electron basis set is introduced as in Eq. (102) and the numbers of a- and / -electrons, Na and N13, respectively, are replaced by sums over diagonal matrix elements Y (P"S)W and E (P S) w [cf. Eq. (104)], M... 

The gauge potentials and are 2x2 Hermitian traceless matrices, and the Higgs fields and % are also 2x2 matrices. These expectations are real-valued, and the nonzero contributions of the Higgs field on the physical vacuum are given by the diagonal matrix entries [95] ... 

The diagonal matrix element of r2 is the expectation value of (r2), given by5... 

The diagonal matrix elements between half-filled shell states are now considered. If it is assumed that the interaction operator is symmetric under time reversal (also as in case 2), then thm = +1. The diagonal interaction elements are just the expectation value of in the closed shell, which is zero if is not totally symmetric under spatial operations (Another way of saying this is that (H%) vanishes if Mr K X K KMKr ), but obviously has time reversal parity +1. It now follows that if the above criteria are met then the diagonal matrix elements must vanish. 

It appears, therefore, that it is possible to obtain accurate expectation values of the spin-orbit operators for diatomic molecules. Matcha et a/.112-115 have provided general expressions for the integrals involved and from their work Hall, Walker, and Richards116 derived the diagonal one-centre matrix elements of the spin-other-orbit operator for linear molecules. Provided good Hartree-Fock wavefunctions are available, these should be sufficient for most calculations involving diatomic molecules. 

Within the SCF HF framework, the energy ej, as a diagonal element of the Hartree-Fock matrix (5.31), can be shown to be equal to the difference between the expectation values of the hamiltonian for the neutral species X (the Hartree-Fock total energy) and for a positive ion X (—i) described by a Slater determinant built on the basis of canonical orbitals identical to those of X (frozen orbital model). This apparently crude description of the X (—i) can be shown to give the lowest energy for the ion. As a result, ei is an approximate estimate of the ionization potential of the i-th electron ... 

The energy expectation value for the crystal field model in the ensemble form is supplemented by the subsidiary conditions that diagonal elements of the density matrix are between 0 and 1, that their sum equals q, and that the matrix is nonnegative. A variational form is then... 

If the Ja quantum number is not well defined, then J2 — J22 = J2X and the unevaluated expectation value of B(R)J2X is implicitly included in the electronic energy. The pair of terms in in Eq. (3.2.19) give rise to off-diagonal matrix elements (A J = 0, AJa = 0, ACl = 1) between different case (c) states, denoted Q.(Ja) or fi(2S+1A) (or Jan, as for the open-core rare earth oxides and halides). The simplicity of HROT in the case (c) basis set exacts a price in the difficulty of evaluating matrix elements of most operators that include Lz, S2, or Sz. 

A,. - diagonal matrix containing the latgest expected change of the reference values, S. ... 

The one-electron density matrix gives rise to others, for example P KK ri-,r i), whose diagonal element gives the probable number of electrons per unit volume (without reference to spin) in the spatial volume element dri at point ri and the spin density matrix Q KK ri r i), whose diagonal element gives the contribution to the expectation value of the total spin z-component, < Sz >, associated with the same volume element. These functions are related to p KK xi-,x i) as follows ... 

As emphasized, one of the advantages of this model is that it provides explicit wavefunctions which can be used in the computation of expectation values for various operators of interest. Due to limitations of space, we cannot reproduce here the complete set of vibrational wavefunctions obtained in the HCN calculation [76]. However, the typical outcome of the algebraic procedure can be outlined. We obtain a polyad of levels labeled by the numbers Vj and Ig of Eq. (4.56). Each polyad contains a number of local states, such as those listed in Eq. (4.57). The numerical diagonalization of the Hamiltonian matrix is performed separately for each polyad. Thus the eigenvectors derived represent the vibrational wavefunctions in the local basis. A possible outcome of the analysis of the HCN molecule could therefore be given by the following sequence of numbers ... 

There still is a point to be discussed the calculation of energy expectation values, within the EH space framework. This can be done, in practice, using Bom-Oppenheimer approximation, defining in this context an electronic Hamilton operator, adopting some diagonal matrix structure and, in addition, supposing the original scalar wavefunction j normalised ... 

However, within this diagonal matrix formalism the statistical form of expectation values is preserved in any circumstance, at least when system s energy is sought. Indeed, if the extended function diagonal form is chosen, then the following sequence of integral expressions can be easily written ... 

Moreover, the presented simple formalism leads towards a diagonal matrix formulation of the energy and other expectation values in computational Quantum Mechanics. The usual quantum mechanical formalism is not at all lost, but... 

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