EDF: Computing electron number probability distribution functions in real space from molecular wave functions

Abstract

Given an N-electron molecule and an exhaustive partition of the real space ( R 3 ) into m arbitrary regions Ω 1 , Ω 2 , … , Ω m ( ⋃ i = 1 m Ω i = R 3 ), the edf program computes all the probabilities P ( n 1 , n 2 , … , n m ) of having exactly n 1 electrons in Ω 1 , n 2 electrons in Ω 2 , … , and n m electrons ( n 1 + n 2 + ⋯ + n m = N ) in Ω m . Each Ω i may correspond to a single basin (atomic domain) or several such basins (functional group). In the later case, each atomic domain must belong to a single Ω i . The program can manage both single- and multi-determinant wave functions which are read in from an aimpac-like wave function description ( .wfn) file (T.A. Keith et al., The AIMPAC95 programs, http://www.chemistry.mcmaster.ca/aimpac, 1995). For multi-determinantal wave functions a generalization of the original .wfn file has been introduced. The new format is completely backwards compatible, adding to the previous structure a description of the configuration interaction (CI) coefficients and the determinants of correlated wave functions. Besides the .wfn file, edf only needs the overlap integrals over all the atomic domains between the molecular orbitals (MO). After the P ( n 1 , n 2 , … , n m ) probabilities are computed, edf obtains from them several magnitudes relevant to chemical bonding theory, such as average electronic populations and localization/delocalization indices. Regarding spin, edf may be used in two ways: with or without a splitting of the P ( n 1 , n 2 , … , n m ) probabilities into α and β spin components. Program summary Program title: edf Catalogue identifier: AEAJ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAJ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 5387 No. of bytes in distributed program, including test data, etc.: 52 381 Distribution format: tar.gz Programming language: Fortran 77 Computer: 2.80 GHz Intel Pentium IV CPU Operating system: GNU/Linux RAM: 55 992 KB Word size: 32 bits Classification: 2.7 External routines: Netlib Nature of problem: Let us have an N-electron molecule and define an exhaustive partition of the physical space into m three-dimensional regions. The edf program computes the probabilities P ( n 1 , n 2 , … , n m ) ≡ P ( { n p } ) of all possible allocations of n 1 electrons to Ω 1 , n 2 electrons to Ω 2 , … , and n m electrons to Ω m , { n p } being integers. Solution method: Let us assume that the N-electron molecular wave function, Ψ ( 1 , N ) , is a linear combination of M Slater determinants, Ψ ( 1 , N ) = ∑ r M C r ψ r ( 1 , N ) . Calling S Ω k r s the overlap matrix over the 3D region Ω k between the (real) molecular spin-orbitals (MSO) in ψ r ( χ 1 r , … χ N r ) and the MSOs in ψ s , ( χ 1 s , … , χ N s ) , edf finds all the P ( { n p } ) 's by solving the linear system (1) ∑ { n p } { ∏ k m t k n k } P ( { n p } ) = ∑ r , s M C r C s det [ ∑ k m t k S Ω k r s ] , where t m = 1 and t 1 , … , t m − 1 are arbitrary real numbers. Restrictions: The number of { n p } sets grows very fast with m and N, so that the dimension of the linear system (1) soon becomes very large. Moreover, the computer time required to obtain the determinants in the second member of Eq. (1) scales quadratically with M. These two facts limit the applicability of the method to relatively small molecules. Unusual features: Most of the real variables are of precision real*16. Running time: 0.030, 2.010, and 0.620 seconds for Test examples 1, 2, and 3, respectively.