Tables for
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 8.7, p. 713

Section 8.7.2. Electron densities and the n-particle wavefunction

P. Coppens,a Z. Sub and P. J. Beckerc

a732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA,bDigital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA, and cEcole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 Châtenay Malabry CEDEX, France

8.7.2. Electron densities and the n-particle wavefunction

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A wavefunction [\psi (1,2,3,\ldots, n)] for a system of n electrons is a function of the 3n space and n spin coordinates of the electrons. The wavefunction must be antisymmetric with respect to the interchange of any two electrons. The most general density function is the n-particle density matrix, of dimension [4n\times 4n], defined as [ \Gamma ^n(1,2,\ldots, n;1^{\prime },2^{\prime },\ldots, n^{\prime })=\psi (1,2,\ldots, n)\psi ^{*}(1^{\prime },2^{\prime },\ldots, n^{\prime }). \eqno (]In this expression, each index represents both the continuous space coordinates and the discontinuous spin coordinates of each of the n particles. Thus, the n-particle density matrix is a representation of the state in a 6n-dimensional coordinate space, and the n-dimensional (discontinuous) spin state.

The pth reduced density matrix can be derived from ([link] by integration over the space and spin coordinates of np particles, [\eqalignno{ &\Gamma ^p(1,2,\ldots, p\semi 1^{\prime },2^{\prime },\ldots, p^{\prime }) \cr &\quad =\left (\,{n \atop p } \,\right) \int \Gamma ^n(1,2,\ldots, p,p+1,\ldots, n\semi 1^{\prime }, 2^{\prime },\ldots, p^{\prime },\cr&\qquad p+1,\ldots, n) {\,{\rm d}}(p+1)\ldots {\,{\rm d}}n. & (}]According to a basic postulate of quantum mechanics, physical properties are represented by their expectation values [\left \langle F\right \rangle ] obtained from the corresponding operator equation, [ \big \langle F\big \rangle =\big \langle \psi |\widehat {F}|\psi \big \rangle \big/\big \langle \psi |\psi \big \rangle, \eqno (]where [\widehat {F}] is a (Hermitian) operator. As almost all operators of interest are one- or two-particle operators, the one- and two-particle matrices [\Gamma ^1(1\,\semi\,1')] and [\Gamma ^2(1,2\,\semi\,1',2')] are of prime interest. The charge density, or one-electron density, can be obtained from [\Gamma ^1(1\,\semi\,1')] by setting 1′ = 1 and integrating over the spin coordinates, i.e. [\eqalignno{ \rho ({\bf r}) &= n\textstyle\int \Gamma ^1(1,1){\,{\rm d}}s_1 \cr &=n\textstyle\int \psi (1,2,\ldots, n)\psi ^{*}(1,2,\ldots, n){\,{\rm d}}s_1{\,{\rm d}}2\ldots {\,{\rm d}}n. & (}]

An electron density ρ(r) that can be represented by an antisymmetric N-electron wave function and is derivable from that wave function through ([link] is called N-representable. As in any real system, the particles will undergo vibrations, so ([link] must be modified to allow for the continuous change in the configuration of the nuclei. The commonly used Born–Oppenheimer approximation assumes that the electrons re­arrange instantaneously in the field of the oscillating nuclei, which leads to the separation [ \psi =\psi _e({\bf r},{\bf R})\cdot \psi _N({\bf R}), \eqno (]where r and R represent the electronic and nuclear coordinates, respectively, and [\psi _e] is the electronic wavefunction, which is a function of both the electronic and nuclear coordinates. The time-averaged, one-electron density <ρ(r)>, which is accessible experimentally through the elastic X-ray scattering experiment, is obtained from the static density by integration over all nuclear configurations: [ \left \langle \rho ({\bf r})\right \rangle =\textstyle\int \rho ({\bf r},{\bf R})P({\bf R}){\,{\rm d}}{\bf R}, \eqno (]where P(R) is the normalized probability distribution function of the nuclear configuration R. The total X-ray scattering can be derived from the two-particle matrix Γ(1, 2, 1′, 2′) through use of the two-particle scattering operator. The total X-ray scattering includes the inelastic incoherent Compton scattering, which is related to the momentum density π(p).

The wavefunction in momentum space, defined by the coordinates [\widehat {{\bf J}}=\widehat p_{j},s_j] of the jth particle (here the caret indicates momentum-space coordinates), is given by the Dirac–Fourier transform of ψ, [\eqalignno{ \widehat {\psi }(\widehat {1},\widehat {2},\ldots, \widehat {n}) &=(2\pi \hbar)^{-3n/2}\textstyle \int \psi (1,2,\ldots, n) \cr& \quad\times\exp \left(-i/\hbar \textstyle\sum \limits _j{\bf p}_j\cdot {\bf r}_j\right){\,{\rm d}}{\bf r}_1{\,{\rm d}}{\bf r}_2\ldots {\,{\rm d}}{\bf r}_n, \cr& &(}]leading, in analogy to ([link], to the one-electron density matrix [ \pi ^1(\widehat {1},\widehat {1}^{\prime })=n\textstyle\int \psi (\widehat {1},\widehat {2},\ldots, \widehat {n})\psi ^{*}(\widehat {1},\widehat {2},\ldots, \widehat {n}){\,{\rm d}}\widehat {2}\ldots {\,{\rm d}}\widehat {n}, \eqno (]and the momentum density [ \pi (p)=\textstyle\int \pi ^1(\widehat {1},\widehat {1}){\,{\rm d}}s_1. \eqno (]The spin density distribution of the electrons, s(r), can be obtained from Γ1(1, 1′) by use of the operator for the z component of the spin angular momentum. If [\eqalignno{ s({\bf r},{\bf r}^{\prime })&=(2M)^{-1}\textstyle\int s_z(1)\Gamma (1,1^{\prime }){\,{\rm d}}s_1, &(\cr s({\bf r})&=s({\bf r},{\bf r),}}]where M, the total magnetization, is the eigenvalue of the operator [ S_z=\textstyle\sum\limits_{i}s_z({i}).]

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