Version 0.2

Yirong Mo and Jiali Gao

Date of this version of module: September 25, 1999

Date of most recent manual update: September 25, 1999

Copyright 1999

BLW is a program for carrying out electronic structure calculations of valence bond configurations that are defined by block-localized orbitals. Combined with the Hartree-Fock energy for the molecular system, electronic delocalization or resonance energies from these configurations can be determined. The BLW program is currently implemented by interfacing with Gaussian94 checkpoint and integral files, from which the electronic primitive integrals are read. The BLW program allows both single-point energy calculation and geometry optimization of the diabatic electronic state, although the latter is performed via a grid search algorithm.

A) J. Chem. Phys. format:

1. Y. Mo, S. D. Peyerimhoff, J. Chem. Phys. 109, 1687 (1998).
2. Y. Mo, Y. Zhang, J. Gao, J. Am. Chem. Soc. 121, 5737 (1999).
3. Y. Mo, and J. Gao, BLW-version 0.2, State University of New York at Buffalo, Buffalo, N.Y., 1999.

B) American Chemical Society format:

1. Mo, Y.; Peyerimhoff, S. D. J. Chem. Phys. 1998, 109, 1687.
2. Mo, Y.; Zhang, Y.; Gao, J. J. Am. Chem. Soc. 1999, 121, 5737.
3. Mo, Y.; Gao, J. BLW-version 0.2, State University of New York at Buffalo, Buffalo, N.Y., 1999.

BLW–version 0.2 is a program that makes use of Gaussian94 output files, including checkpoint and integral files. BLW–version 0.2 will be distributed from the State University of New York at Buffalo (until December 31, 1999, after January 1, 2000, from University of Minnesota). The user will require the g94int utility for converting Gaussian94 files into format that is used by BLW. This utility file g94int is distributed along with BLW. The BLW is not directly interfaced with Gaussian94, though it uses the information and integral files generated by Gaussian94. Thus, the prospective user of BLW should have access to Gaussian94 and should obtain a valid license for Gaussian94 from Gaussian, Inc.

Title page                                                                                     1

BLW abstract                                                                              2

Referencing for BLW                                                                   3

Availability of BLW                                                                      4

Contents                                                                                      5

1.    INTRODUCTION                                                               6

2.    THEORETICAL BACKGROUND                                      6

        2.A. The block-localized wave function (BLW) method

        2.B. Calculation of resonance energy

3.     PROGRAM STRUCTURE                                                  10

        3.A. Overall design

4.     INSTALLATION OF BLW AND ITS USE                          11

        4.A. Program distribution

        4.B. Installation

5.     DESCRIPTION OF INPUT                                                  12

6.     TEST RUN                                                                            13

BLW is a program for carrying out electronic structure calculations of valence bond-like configurations. There is considerable interest in determining the resonance energy of a molecular system. Resonance energy refers to the energy gained from electronic delocalization in a molecular system relative to a well-defined, hypothetical reference state that lacks of such an electronic delocalization. The reference state corresponds to a localized diabatic electronic configuration, which may be related to a particular Lewis resonance structure or a valence bond configuration. The best known example perhaps is the resonance energy or aromaticity of benzene, where the aromaticity is defined as the energy difference between benzene and that of 1,3,5-cyclohexatriene. Therefore, to evaluate the resonance energy of a molecular system, it is essential to be able to define the corresponding localized diabatic state. The BLW program is designed to define such diabatic states and to determine their electronic energies.

The block-localized wave function (BLW) method is developed to circumvent the delocalized nature of molecular orbitals in Hatree-Fock (HF) theory to study properties of localized, or valence bond-like, electronic structures. Although ab initio valence bond (VB) method can be used to study resonance effect and to define electronic localized states, its computational costs can quickly become intractable and thus prevent applications to large molecular systems. The BLW method provides a convenience approach to define valence bond-like resonance configurations at the computational cost comparable to HF molecular orbital calculations.

Before we proceed, it should be pointed out that the localized molecular orbitals derived from unitary transformation of canonical orbitals are in fact not strictly localized. These "localized" molecular orbitals contain both orthogonalization and delocalization tails, the latter of which makes contribution to the electronic delocalization effect and is not appropriate for describing diabatic potential energy surfaces. To avoid these two types of tails in the BLW method, we impose restrictions on the expansion space of the molecular orbitals.

We start by noting the fact that it is always possible to uniquely partition all electrons and basis orbitals of a molecular system into k subgroups, corresponding to a particular form of the Lewis resonance or VB structure. For simplicity, we assume that each subgroup is a closed shell system; the ath subgroup contains m_a basis functions and n_a electrons. The extension of the BLW method to open shell systems is straightforward. Thus, the total number of primitive basis functions, M, and the total number of electrons, N, in the system are:

and                                                                                 (1)

Each molecular orbital in a subgroup a is written as a linear combination of the primitive basis functions {c^a_m, m = 1, ..., m_a} restricted in that particular subspace.

                                                                                                   (2)

The Slater determinant wave function for resonance structure, s, is then constructed with the use of these occupied molecular orbitals in the system:

                                                                                         (3)

where is an antisymmetrizing operator, and F_a is a successive product of the occupied MOs in the ath subgroup (eq 2).

                                                                                             (4)

It is important to note that the wave function defined in eq 3 is subjected to the restriction that molecular orbitals within each subgroup are orthogonal, whereas orbitals between different subgroups have non-zero overlap - a feature of the valence bond approach:

                                                                                         (5)

where O_ij is the overlap integral between MOs i and j in different subgroups. With the definition of eqs 2 and 3, the coefficient matrix for the occupied MOs of the BLW wave function has the following form:

                                                                                         (6)

where the element C_a is an s_a X n_a/2 matrix. The energy of the localized wave function is determined as the expectation value of the Hamiltonian H, which is giving as follows:

                                         (7)

In eq 7, h_mn and F_mn are, respectively, elements of the usual one-electron and the Fock matrices, and d_mn is an element of the density matrix, D, which is evaluated by using eq 8.

                                                                                                     (8)

where S is the overlap matrix of the basis functions.

The molecular orbitals in the BLW wave function can be optimized in two ways: (1) the Jacobi rotation method, and (2) the reorthogonalization method. Both approaches have been implemented in the BLW program. The reorthogonalization method is approximately 10 times faster than the successive Jacobi rotation algorithm and it is now the default option in BLW. The Jacobi rotation algorithm was used and describe in (Y. Mo, S. D. Peyerimhoff, J. Chem. Phys. 1998, 109, 1687).

With the construction of a block-localized wavefunction, the delocalization energy (DE_d) can be conveniently defined as follows:

                                                                         (9)

where E[Y(HF)] and E[Y(BLW)] are energies determined, respectively, using the HF wave function which corresponds to the delocalized state and using the BLW wave function which is defined for a particular reference diabatic state.

If the diabatic state is properly chosen, eq 9 can be conveniently used to estimate energies associated with p conjugation, s-p hyperconjugation, and n-s* negative hyperconjugation. A recent application of the BLW method to investigate the n-s* negative hyperconjugation and p-d bonding was reported in (Y. Mo, Y. Zhang, J. Gao, J. Am. Chem. Soc. 1999, 121, 3757).

.

The BLW program makes use of the electronic integrals computed by Gaussian94 and carries out BLW-SCF electronic structure calculations. There is no direct interface between the BLW program and Gaussian94, although it uses the Gaussian utility program formchk and another utility program g94int that translates Gaussian94 format into that of BLW format. These utility programs are used primarily for convenience in other applications. The BLW method has also been implemented into a locally modified version of the public domain program GAMESS, and has been incorporated into the MCQUB program for statistical mechanical Monte Carlo simulations of condensed phase systems. Here, these utility programs are completed avoided.

BLW–version 0.2 is distributed by the State University of New York at Buffalo (after January 1, 2000, by the University of Minnesota), and the utility program g94int is also distributed along with BLW (gao@chem.umn.edu or ymo@chem.umn.edu). To successfully execute BLW, the user must have access of Gaussian94 and its accompanying utility programs. Gaussian94 is a licensed program by Gaussian, Inc.

The user should carry out the following steps:

1. Obtain the BLW and the g94int utility program (gao@chem.umn.edu or ymo@chem.umn.edu). Compile blw.f using your FORTRAN compiler, e.g., "f77 -o blw -[optimization options] blw.f"
2. Obtain the Gaussian programs. The formchk utility program should have been automatically generated if Gaussian94 has been successfully compiled.
3. To generate the g94int utility program, first, define the path for the Gaussian94 util.a library in makefile. Then, run the make file: "make -f makefile".

BLW is run using a single script file which (1) execute the Gaussian calculations and generate the necessary integrals, (2) reformat the Gaussian files, and (3) run BLW calculations. Examples are provided in the test jobs.

The "card" input for the BLW calculation (see test1.inp) is described below:

Line 2i: NEBi, NOBi, IOPi (where i is the ith repeat of line 2 and 3).

Line 3i: O(1), O(2), ..., O(NOBi)

This section describes one test job. It includes a full input file, initial coordinates that has been previously optimized, and an output file that can be checked against the user's output so that he or she will be confident that the code has performed correctly on his or her machine.

This test job (1) performs a single-point HF/6-31G* calculation, (2) reformat the Gaussian integrals for BLW, and (3) performs BLW calculation for a block localized configuration that includes two blocks, one containing the lone pair electrons on nitrogen, and the other consisting of the rest of the electrons. It takes roughly 10 minutes on an SGI Origin 200 computer (175 MHz).

6.1A. Input files

The test1.com file contains the script commands for executing the entire test job calculations. It generates a number of temporary scratch files and removes them after the job is completed. The Gaussian output is written into test1.out, and BLW output into test1blw.out.

The test1.inp file contains the instruction commands for the BLW program to define the "block" electronic structure in terms of the orbitals and electrons included in each block. Obviously, the user should verify that the MO's included in each block have the correct symmetry and of course ALL orbitals with the required symmetry have been included.