Electron-pair uncertainty relationships and the intracule–extracule isomorphism

Electron-pair uncertainty relationships of generalized Heisenberg type in quantum mechanics are formulated. They express restrictions imposed by quantum theory on the electron-pair intracule (relative motion) and extracule (centre-of-mass motion) densities in the two complementary spaces by means of the corresponding statistical moments. These relations are shown to reduce to an elementary or Heisenberg form when the intracule–extracule isomorphism holds. The validity of this isomorphism is also discussed for some special systems. The quantum position–momentum uncertainty principle for an N -electron system characterized by the position (momentum) wavefunction (x1, . . . ,xN) ( (y1, . . . ,yN)), where xk ≡ (rk, σk) and yk ≡ (pk, σk) are the combined position-spin and momentumspin coordinates of the kth electron respectively, was originally expressed by the Heisenberg inequality [1] 〈r2〉〈p2〉 9 4N. (1) Atomic units are used throughout this letter. The position (momentum) expectation value 〈r2〉 (〈p2〉), which is the second central moment of the one-electron density function ρ1(r) ( 1(p)), gives an uncertainty measure of the position (momentum) of a single electron in the system. The joint position–momentum uncertainty measure is, according to (1), bounded from below by the number of particles of the system. Later on, this principle was expressed in various generalized and improved forms by the use of one-electron statistical moments rather than the second and/or information entropies to describe the single-electron position and momentum uncertainties [2–4]. Let us highlight the general one-electron uncertainty relationship [3] given by 〈rm〉〈pm〉 CmN, m > 0 (2) 0953-4075/02/130309+06$30.00 © 2002 IOP Publishing Ltd Printed in the UK L309 L310 Letter to the Editor and Cm ≡ 9em−2 m2 [ 9π 16 2(1+3/m) ]m/3, where (x) is the Euler gamma function and e = 2.718 . . . is the exponential number, which expresses indeterminacy of positions and momenta in terms of the mth moments of the corresponding distributions, and is included in equation (1) as a particular case. To properly study the correlation effects in atoms and molecules due to the electron– electron repulsion term of the corresponding Hamiltonian, one has to go beyond the oneelectron density ρ1(r), which considers the interelectronic interaction only in an average sense, and to work with the position (momentum) electron-pair density function ρ2(r1, r2) ( 2(p1,p2)) given by ρ2(r1, r2) = N(N − 1) 2 ∫ dσ1 dσ2 . . . dσN dr3 . . . drN | (r1, σ1, . . . , ri , σi, . . . , rN, σN)|. The use of the relative and centre-of-mass coordinates to describe the motion of the two interacting particles allows one [5] to decompose the six-variable function ρ2(r1, r2) into two three-variable functions, the electron-pair intracule (relative motion) density ρI(u) = ∫ dr1 dr2 δ(u− (r1 − r2))ρ2(r1, r2), (3) and the electron-pair extracule (centre-of-mass motion) density ρE(R) = ∫ dr1 dr2 δ(R− (r1 + r2)/2)ρ2(r1, r2), (4) where δ(r) is the three-dimensional Dirac delta function. The intracule density ρI(u) describes the probability density function for the relative vector rk−rl of any pair of electrons k and l as u, and the extracule ρE(R) represents the probability density function for the centre-of-mass vector (rk + rl)/2 of any pair of electrons k and l as R. An exactly analogous decomposition can be written for the electron-pair density function in momentum space, 2(p1,p2), by means of the momentum electron intracule πI(v) and extracule πE(P ) densities, which represent the electron-pair density functions for the relative momentum vector pk − pl as v and for the centre-of-mass momentum vector (pk + pl)/2 as P , respectively. Nowadays it is becoming more and more apparent that numerous aspects of the electron correlation problems in atoms and molecules can be explicitly and easily interpreted in terms of these position and momentum intracule and extracule densities. For example, intracule and extracule moments in position space (〈un〉 = ∫ ρI(u)u du and 〈Rn〉 = ∫ ρE(R)R dR, respectively) represent location and energetic properties of electron pairs in a many-electron system. See [6–8] for a review; we refer to [8] for notations and definitions. To date there does not exist in quantum mechanics any general or elementary electron-pair uncertainty principle connecting the intracule and extracule electron-pair moments of equal order in position and momentum spaces. The main purpose of this letter is to formulate such a general electron-pair uncertainty principle, and to study its reduction to an elementary or Heisenberg form by use of an isomorphism between the intracule and extracule properties of the N -electron system under consideration. However, the following rigorous rule, recently found by Koga [9] is known: 〈u2〉 + 4〈R2〉 = 2(N − 1)〈r2〉, (5) which connects the position one-electron and two-electron intracule and extracule moments. An exactly analogous expression also holds for the corresponding momentum oneand twoelectron moments 〈p2〉, 〈v2〉 and 〈P 2〉. To obtain equation (5), Koga introduced a generalized electron-pair density function g(q; a, b) defined by g(q; a, b) = (4πq2)−1 〈N−1 ∑