Data processing: structural design – modeling – simulation – and em – Simulating nonelectrical device or system – Chemical
Reexamination Certificate
2000-04-07
2004-09-28
Thomson, W (Department: 2123)
Data processing: structural design, modeling, simulation, and em
Simulating nonelectrical device or system
Chemical
C703S011000, C703S002000, C708S446000
Reexamination Certificate
active
06799151
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a method and apparatus for parallel processing preferably used for calculation of massive matrix elements having certain symmetry and, more particularly, for processing calculation of Fock matrix elements at a high speed during simulation of a molecule using the ab initio molecular orbital method.
2. Description of the Related Art
Techniques for numerical analysis of the state and behavior of molecules in the field of chemistry include the molecular orbital method, molecular dynamics method and Monte Carlo method. Among such methods, ab initio molecular orbital calculation are quantum-mechanical calculations based on the first principle which are aimed at describing the behavior of electrons in a molecule. Therefore, this method is regarded as a basis for simulation of a molecule and is a technique of importance from the industrial point of view which is used for close analysis of structures of substances and chemical reactions.
According to the ab initio molecular orbital calculation, a basis function is the inverse of an exponential function whose exponent is obtained by multiplying an empirical coefficient by the square of the distance between the atomic nucleus of an atom forming a part of a molecule and an orbital electron or linear combination of them, and a plurality of such basis functions are prepared for one atom. A wave function of an orbital electron in a molecule, i.e., a molecular orbital is described by linearly combining those basis functions.
A primary process of the ab initio molecular orbital calculation is to determine linear combination coefficients for the basis functions of molecular orbitals, and the calculation necessitates computational complexity and storage capacity proportion to the biquadratic of the number of the basis functions. For this reason, ab initio molecular orbital calculations are only used for molecular systems on the scale of 100 atoms or so at present. In order to realize more practical analysis of biological and chemical phenomena from the viewpoint of molecular theory, it is essential to develop a calculation system dedicated to ab initio molecular orbital calculations which is applicable even to molecular systems on the order of several thousand of atoms.
[Summary of Ab initio Molecular Orbital Calculation]
In the ab initio molecular orbital calculation, a state of a molecule &psgr; is described using an electron orbital &phgr;
&mgr;
which corresponds to a spatial orbital of an electron in the molecule. Here, &mgr; is a suffix which indicates a &mgr;-th orbital among plural molecular orbitals. A molecular orbital &phgr;
&mgr;
is a linear combination of atomic orbitals &khgr;
I
which is approximated by Expression 1 in FIG.
23
.
In Expression 1, I is a suffix which indicates an I-th atomic orbital among plural atomic orbitals. An atomic orbital is also referred to as “basis function”. In this specification, an atomic orbital is hereinafter referred to as “basis function”. C
I
&mgr;
in Expression 1 is a linear combination coefficient. The sum regarding I in Expression 1 is related to all basis functions that form a molecule to be calculated.
In order to describe molecular orbitals on a quantum mechanical basis, the states of electrons in the molecule must satisfy the well known Pauli exclusion principle. A Slater determinant like Expression 2 in
FIG. 23
is used as an expression to describe a state &psgr; of the molecule with 2n electrons such that it satisfy the Pauli exclusion principle taking electron spins into consideration. &agr;(x) and &bgr;(x) in Expression 2 respectively represent states in which an x-th electron spin is upward and downward.
The Hamiltonian H for the molecule with 2n electrons is in the form of the sum of a one-electron part H
1
and two-electron part H
2
and is expressed by Expressions 3 through 5.
The first term in the parenthesis on the right side of Expression 4 in
FIG. 23
represents a kinetic energy of electrons p, and the second term represents an interaction between a p-th electron and an A-th atomic nucleus. In Expression 4, &Sgr;
p
(&Sgr;
i
represents sum regarding i throughout the present specification) represents the sum of all electrons; &Sgr;
A
represents the sum of all atomic nuclei; Z
A
represents the charge of an atomic nucleus A; and r
p
A
represents the distance between an electron p and the atomic nucleus A.
Expression 5 represents an interaction between electrons p and q. &Sgr;
p
&Sgr;
q(>p)
represents a sum of combinations of two electrons; and r
pq
represents the distance between electrons p and q.
By using the above-described Hamiltonian H and the Slater determinant in Expression 2, an expected value &egr; of a molecular energy is expressed by Expressions 6 through 9 in FIG.
24
.
In Expression 6, &Sgr;
&mgr;
and &Sgr;
&ngr;
represent a sum regarding n (n is a positive integer) molecular orbitals. Expression 7 is referred to as “core integral” and describes a typical electron which is numbered “1”. Expressions 8 and 9 are respectively referred to as “Coulomb integral” and “exchange integral” and describe typical electrons
1
and
2
.
Expression 6 can be changed using basis functions into Expressions 10 through 13 shown in FIG.
25
. The integral expressed by Expression 13 is referred to as “electron-electron repulsion integral” or simply “electron repulsion integral”.
An expected value &egr; of a molecular energy expressed by Expression 10 includes unknown coefficients C
I
&mgr;
, and no numerical value is obtained as it is. C
I
&mgr;
correspond to the linear combination coefficients in Expression 1, where &mgr; represents integers 1 through n (the number of molecular orbitals); I represents integers 1 through N (N is the number of basis functions which is a positive integer). Hereinafter, an N×n matrix C whose elements are C
I
&mgr;
is referred to as “coefficient matrix”.
One method used to determine a coefficient matrix that minimizes the expected value &egr; to obtain a wave function &psgr; in a ground state is the Hartree-Fock-Roothaan variational method (hereinafter referred to as “HFR method”. Expressions 14 through 18 in
FIG. 26
are Expressions obtained by the HFR method, although the process of derivation is omitted.
F
IJ
and P
KL
are respectively referred to as “Fock matrix elements” and “density matrix elements”. In the following description, those elements are sometimes expressed by F(I, J) and P(K, L). They have numerical values corresponding to I and J or K and L which are integers 1 through N, and are each expressed in the form of an N×N matrix.
A coefficient matrix is obtained by solving Expression 14. Expression 14 is a system of n×N expressions because it applies to all &mgr;'s from 1 through n and all I's from 1 through N.
A density matrix P is used to calculate a coefficient matrix C obtained by solving Expression 14. A density matrix P is calculated from a coefficient matrix C as indicated by Expression 18. Referring to the calculation procedure specifically, an appropriate coefficient matrix C is initially provided; a Fock matrix F is calculated from Expression 15 using a density matrix P calculated using the same; and a new coefficient matrix C is obtained by solving the simultaneous equations of Expression 14. The above-described calculations are iterated until the difference between the primitive coefficient matrix C from which the density matrix P is obtained and the resultant coefficient matrix C becomes sufficiently small, i.e., until it becomes self-consistent. Such iterative calculations are referred to as “self-consistent field calculation” (hereinafter abbreviated to read as “SCF calculation”).
The most time-consuming calculation in practice is the calculation of Fock matrix elements F
IJ
of Expression 15. The reason is that Expression 15 must be calculated for each combination of I and J and that the sum regarding K and L of the density matrix elements P
KL
must be calculated for each combination of I
Inabata Shinjiro
Kitamura Kunihiro
Miyakawa Nobuaki
Obara Shigeru
Takashima Hajime
Taisho Pharmaceutical Co., Ltd
Thomson W
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