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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

The density matrix in many-electron quantum mechanics II. Separation of space and spin variables; spin coupling problems

R. McWeeny, Y. Mizuno

Published Jan 24, 1961• 214 citations
Publisher

Abstract

Abstract The spin-independent properties of an electronic system are completely determined by spinless density functions, P1(1; 1') and P 2( 1,2; 1',2 '), derived from the 1-and 2-electron density matrices, p1(1; 1') and p2(1,2 ; 1',2'). These functions, and their physical inter­-pretation, have been dealt with in part I: the present paper contains the generalizations necessary when the Hamiltonian contains spin terms (as in discussions of electron and nuclear spin resonance). The density matrices, and more generally the transition density matrices p1(kλ|1; 1') and p2(kλ|l,2; 1',2') associated with states and Φkand Φλ are resolved into spinless components associated with different spin situations: p1 has four such components (one giving the density of a-spin electrons, a second the density of β-spin electrons, and two with less direct physical significance), while p2 has sixteen components. It is shown that all components, for states k=(S, M),λ= (S',M') of given S, S', are determined by three 2-electron functions.For k=λ these reduce to p2, the spinless pair function, from which the electron density p1 may be derived by integration; Q2a ‘conditional' spin density, from which the (1-electron) spin density Q1 may be derived (as in the case of p1 and P2); Q c. a., a ‘coupling anisotropy function', which describes the anisotropy of spin-spin coupling associated with quantization of the spin z-component. It is also useful to introduce (though it is not an independent function). Qc, a ‘coupling density function', which measures the coupling between the spins of two electrons at different points in space. All spin-dependent effects, including the usual spin-spin and spin-orbit couplings and also (indirect) nuclear and electron-nuclear spin-spin couplings, may be related to the many- electron wave function through the above density functions. All the densities have a physical interpretation, independent of the precise nature of the wave functions (which may, for example, be the exact eigenfunctions of a spinless Hamiltonian or merely approximations of orbital form). The derivation of the density functions from approximate wave functions is considered in some detail. For functions describing weakly coupled electron groups (e. g. inner-shell and valence electrons) the densities may be expressed in terms of those of the separate groups. This reduction may be followed by an orbital description of each group; and subsequently, by a linear-combination-of-atomic-orbitals (l. c. a. o.) approximation. In the simpler approxi­mations, all the coupling effects discussed are determined by electron and spin ‘populations'. Preliminary calculations suggest that such approximations may in many cases be adequate.

DOI
10.1098/rspa.1961.0008

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Abstract The spin-independent properties of an electronic system are completely determined by spinless density functions, P1(1; 1') and P 2( 1,2; 1',2 '), derived from the 1-and 2-electron density matrices, p1(1; 1') and p2(1,2 ; 1',2'). These functions, and their physical inter­-pretation, have been dealt with in part I: the present paper contains the generalizations necessary when the Hamiltonian contains spin terms (as in discussions of electron and nuclear spin resonance). The density matrices, and more generally the transition density matrices p1(kλ|1; 1') and p2(kλ|l,2; 1',2') associated with states and Φkand Φλ are resolved into spinless components associated with different spin situations: p1 has four such components (one giving the density of a-spin electrons, a second the density of β-spin electrons, and two with less direct physical significance), while p2 has sixteen components. It is shown that all components, for states k=(S, M),λ= (S',M') of given S, S', are determined by three 2-electron functions.For k=λ these reduce to p2, the spinless pair function, from which the electron density p1 may be derived by integration; Q2a ‘conditional' spin density, from which the (1-electron) spin density Q1 may be derived (as in the case of p1 and P2); Q c. a., a ‘coupling anisotropy function', which describes the anisotropy of spin-spin coupling associated with quantization of the spin z-component. It is also useful to introduce (though it is not an independent function). Qc, a ‘coupling density function', which measures the coupling between the spins of two electrons at different points in space. All spin-dependent effects, including the usual spin-spin and spin-orbit couplings and also (indirect) nuclear and electron-nuclear spin-spin couplings, may be related to the many- electron wave function through the above density functions. All the densities have a physical interpretation, independent of the precise nature of the wave functions (which may, for example, be the exact eigenfunctions of a spinless Hamiltonian or merely approximations of orbital form). The derivation of the density functions from approximate wave functions is considered in some detail. For functions describing weakly coupled electron groups (e. g. inner-shell and valence electrons) the densities may be expressed in terms of those of the separate groups. This reduction may be followed by an orbital description of each group; and subsequently, by a linear-combination-of-atomic-orbitals (l. c. a. o.) approximation. In the simpler approxi­mations, all the coupling effects discussed are determined by electron and spin ‘populations'. Preliminary calculations suggest that such approximations may in many cases be adequate.
The density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)
Many-electron theory is treated here with much attention focused on the synthesis of the density functional and density matrix theory. This synthesis shows that the exchange–correlation potential, normally obtained as the functional derivative with respect to the electron density of an assumed approximate exchange–correlation energy, can be written exactly in terms of the “near-diagonal” behavior of the fully interacting first-order density matrix and the diagonal of the second-order density matrix, thereby bypassing the functional derivative. Also involved, since the exchange–correlation potential is an “artifact” of reducing the many-electron Schrödinger equation to one-body Slater–Kohn–Sham equations, are the one-body orbitals derived from these latter equations, not just the density thereby obtained. This situation has then motivated a discussion of what constraints are placed on the low-order density matrices of atoms and molecules, described by an assumed spin-independent Hamiltonian, by the requirement that these matrices are constructed from an N-electron wave function which is an eigenfunction of the spin operators S2 and Sz. Some briefer discussion is added on the optimally convergent determinantal expansion of such an N-electron wave function, the relation between Brueckner and Löwdin natural orbitals being commented on as a by-product. In this latter context, it is noted that the exchange-only potential of density functional theory can be represented as a sum of two line integrals, each of which is path-dependent, the sum being, of course, path-independent. But in addition to knowledge of Slater–Kohn–Sham orbitals appearing as in the Harbola–Sahni exchange-only potential (which is one of the path-dependent terms), there is a kinetic correction. This involves Löwdin natural orbitals and their occupation numbers, but only now in the weak coupling limit of the electron–electron interaction, as emphasized in the kinetic “correction” to the Harbola–Sahni formula exhibited by Levy and the writer. Some directions for further work are briefly set out. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 163–176, 1999
In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of NNN antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order kkk has two sets of indices of each kkk variables, and it is further antisymmetric in each set of these indices.Every normalizable antisymmetric wave function may be expanded in a series of determinants of order NNN over all ordered configurations formed from a basic complete set of one-particle functions ${\ensuremath{\psi}}_{k}$, which gives a representation of the wave function and its density matrices also in the discrete kkk-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly NNN spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence.In case the basic set is of finite order MMM, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.
A fully spin-adapted approach to many-electron density matrices is developed in the context of the unitary group approach to many-electron systems. An explicit expression for the single-electron spin-density operator, as a polynomial of degree two in the orbital U(n) generators, is derived for the case of spin-independent systems. Extensions to spin-dependent systems are also considered, leading to the appearance of total-spin transition densities, whose general properties are investigated. A corresponding formalism for the two-electron density matrix, which is capable of further generalization, is also developed. The results of this paper, together with recent developments on the matrix elements of the U(2n) generators in the electronic Gel’fand basis, afford a versatile method for the direct calculation of one- and two-body density matrices in the unitary group approach framework.
The formulation of the density matrix functional theory (DMFT) in which the correlation component U of the electron–electron repulsion energy is expressed in terms of a model two-electron density cumulant matrix (a.k.a. the 2-cumulant) and the tensor of two-electron integrals g is critically analyzed. The dependence of on both the vector of occupation numbers n and g derived from the corresponding vector (1) of natural spinorbitals, and its concomitant invariance to the nature of the spin-independent interparticle interaction potential that enters the definition of g are emphasized. In the case of spin-polarized systems, (and thus U ) is found to be a function of not only n and g, but also of the vector and the matrix, where is the spin-flip operator. The presence of spin polarization imposes additional constraints upon, including a simple condition that, when satisfied, assures the underlying wavefunction being an eigenstate of both the and operators with the eigenvalues and, respectively. This feature allows targeting electronic states with definite multiplicities, which is virtually impossible in the case of the Kohn–Sham implementation of density functional theory. Among the four possible pairing schemes for the natural spinorbitals that give rise to approximations employing with only two independent indices, three are found to result in unphysical constraints even for spin-unpolarized systems, whereas the failure of the fourth one turns out to be precipitated by the presence of spin polarization. Consequently, any implementation of DMFT based upon “two-index” is shown to be generally unsuitable for spin-polarized systems (and incapable of yielding the spin-parallel components of U for the spin-unpolarized ones). A clear distinction is made between the genuine 1-matrix functionals that are defined for arbitrary N -representable 1-matrices and general energy expressions that depend on auxiliary quantities playing the role of fictitious 1-matrices subject to additional (often unphysical) constraints.
When we treat an atom containing a number n of electrons by the method of the self-consistent field, we assume that each electron has its own particular “orbit,” specified by a wave function ( q | r ) in four variables q . These four variables are usually taken to be the three coordinates of the electron together with a variable describing the spin, but according to the transformation theory of quantum mechanics, they may be any four independent commuting functions of the coordinates, momenta and spin variables.