A new measure of electron correlation

We propose to quantify the “correlation” inherent in a many-electron (or manyfermion) wavefunction ψ by comparing it to the unique uncorrelated state that has the same 1-particle density operator as does |ψ〉〈ψ|. Electron correlation is of fundamental importance in quantum chemistry and magnetism, but as yet there is no definitive way to quantify it: given the wavefunction ψ(x1, x2, . . . , xN) representing the state of a system of N electrons, how much “correlation” is there in that N -electron state? By definition, a wavefunction that has the form of a Slater determinant represents an “uncorrelated” state, but how much “correlation” should be attributed to states that are not represented by a Slater determinant wavefunction? Some measures of “correlation” have already been advanced in the literature: the “degree of correlation” [3, 4] and the “correlation entropy” [5, 6, 7, 8], for example. These correlation measures depend completely upon the eigenvalues of the 1-particle statistical operator Γ, i.e., the operator with the integral kernel γ(x, y) = N ∫ ψ(x, z2, . . . , zN)ψ(y, z2, . . . , zN) μ(dz2) · · ·μ(dzN) (1) (supposing that the wavefunction ψ has norm 1, so that the trace of Γ equals the number of electrons). The “degree of correlation” [3, 4] is inversely proportional to the sum of the squares of the eigenvalues of Γ, and the “correlation entropy” [5, 6, 7, 8] is proportional to the von Neumann entropy of Γ. Such measures ascribe the same amount of “correlation” to all wavefunctions that have the same 1-particle statistical operator. We feel it a severe conceptual limitation to have to say that all wavefunctions that have the same 1-particle operator contain the same amount of correlation, and we hereby propose a new measure of correlation that does not suffer that limitation. Wolfgang Pauli Institute, Nordbergstr. 15, A–1090 Wien, Austria ([email protected]). WPI c/o Fak. f. Math., Univ. Wien, Nordbergstr. 15, A–1090 Wien, ([email protected]). 1 States represented by Slater determinant wavefunctions are the only pure states that we deem “uncorrelated”; but we also recognize that certain mixed states should also be regarded as “uncorrelated”, namely, mixed “quasifree states”. These have random particle number and must be represented by density operators (i.e., statistical operators of trace 1) on the fermion Fock space. Let H = L(R) be the 1-electron Hilbert space, and let af and af denote the electron creation and annihilation operators for f ∈ H. A density operator IΓ on the fermion Fock space over H represents a “quasifree state” if Tr ( IΓafn · · ·a † f2 a † f1 ag1ag2 · · ·agn ) = det ( Tr(IΓafiagj) n i,j=1 (2) for all n and all f1, g1, . . . , fn, gn ∈ H. The two-point correlations Tr(IΓafag) determine all higher correlations under the quasifree state IΓ, and in this sense a density operator satisfying (2) may be called “uncorrelated”. The 1-particle operator Γ defined by 〈g,Γf〉 = Tr(IΓafag) (3) has trace equal to the average particle number under IΓ, i.e., Tr(Γ) = Tr(IΓN) where N denotes the number operator. In case the average particle number is finite, we can reconstruct IΓ from Γ as follows. Let {φi}i=1 be a complete system of eigenvectors of Γ with Γ(φi) = λiφi . (4) The eigenvalues λi are all between 0 and 1; we will interpret them as “occupation probabilities”. Let SS denote the class of all finite sets of positive integers, including the empty set. We can choose a random member of SS, that is, we can form a random set s of positive integers, by including i in s with probability λi (and excluding it with probability 1− λi) independently of all other positive integers which may or may not be included in s. This procedure produces the set s with probability p(s) = ∏