Parallel computer system and parallel computing method

Details Parallel computer system and parallel computing method Parallel computer system and parallel computing method

2000-04-06

2003-10-07

Ngo, Chuong Dinh (Department: 2124)

Electrical computers: arithmetic processing and calculating

Electrical digital calculating computer

Particular function performed

C708S501000

Reexamination Certificate

active

06631391

ABSTRACT:

BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a parallel computer system for use in analysis of physical phenomena containing a large number of high precision sum of products operations by way of floating-point arithmetic operation including non-empirical calculation of molecular orbit used in designing the molecular structure of medicines and in prediction of physical properties.
2. Description of the Related Art
Along with the micronization and the increase of speed of semiconductor devices, high performance computers have come to be realized and molecular simulations using non-empirical calculation of molecular orbits have come to be conducted also in the fields of design of molecular structure and of prediction of values of physical properties in pharmacology.
Among the non-empirical calculations of molecular orbits, Hartree-Fock (HF) method which requires a relatively less amount of calculation and which can fully accommodate with qualitative analysis has been most widely used. The HF method has been described in Shigeru Fujinaga, “Molecular Orbit Method” Iwanami Shoten, 1980, in Eiji Osawa, “Molecular Orbital Method”, Kodansha Scientific, 1994, and in Osamu Kikuchi, “Basics of Quantum Chemistry” Asakura Shoten, 1997 for example. The outline of the HF method will be described below.
The HF method is formulated as a method for solving Fock equation by a SCF method described later. Here, the Fock equation may be expressed by the following expression which is obtained as a result of implementing one-electron approximation or linear approximation to Schroedinger equation for the whole molecule:
FC=SC &egr;
  (1)
where, N is a total number of atomic orbits contained in the molecule and m is a total number of molecular orbits expressed by linear approximation of the atomic orbits. Because energy of the molecule may be found by solving this Fock equation, it is possible to judge whether the molecule is stable or not by its value.
In the equation (1), F is a matrix of N×N called a Fock matrix, S is a matrix of N×N called an overlapping matrix, C is a matrix of N×m representing a coefficient and &egr; is a diagonal matrix of m×m representing energy of each electron occupying the molecular orbit.
Here, an element Frs (r, s=1 to N) of the Fock matrix may be expressed by the following equation:
Frs=hrs+grs=hrs+&Sgr;[t, u=
1~
N]Ptu
((
rs, tu
) −(½)(
rt, su
))  (2)
hrs in this equation (2) is an integral amount representing energy to one electron and is calculated by a number proportional to N
2
in one time of calculation of the equation (1).
It is noted that in this specification, &Sgr;[i, j=1 to N] f(i, j) indicates an arithmetic operation for finding a total sum from 1 to N for i and j with respect to a function f(i, j). &Sgr;[i =1 to N]f(i) indicates an arithmetic operation for finding a total sum from
1
to N for i with respect to a function f(i).
Ptu in the equation (2) is called a density matrix and is expressed as follows by using the above-mentioned matrix C:
Ptu
=&Sgr;(
j=
1~
m
)
Cfj·Cuj
  (3)
(rs, tu) (r, s, t, u=1 to N) in the equation (2) is a physical amount called two-electron integral and is represented as follows by using the atomic orbit x
i
(r)(i=1 to N, r is a coordinate):
(
rs, tu
)=∫∫
xr
(
r
1
)
xs
(
r
1
)(1
/r
12
)×
xt
(
r
2
)
xu
(
r
2
)
dr
1
·
dr
2
  (4)
Here, r
1
and r
2
are coordinate systems independent from each other and double integration is carried out across the respective whole spaces. r
12
represents a distance between the coordinate systems r
1
and r
2
. Because r, s, t and u exist by the number of atomic orbits, respectively, the two-electron integral is required by a number proportional to N
4
in one time of calculation of the equation (1).
The element Srs of the overlapping matrix S may be represented by the following equation:
Srs=∫xr
(
r
1
)
xs
(
r
1
)
dr
1
  (5)
Because the HF method is represented as described above, it is a question of finding m characteristic values &egr;i and characteristic vector Ci (i=1 to m) represented by the equation (1). However, as it is apparent from the equations (2) and (3), because the Fock matrix contained in the equation (1) may be found by using the vector Ci representing a coefficient, the value of F cannot be found unless Ci obtained by solving the equation (1) is used.
Accordingly, new Ci is found by setting an adequate value as the initial guess of Ci at first, by finding F by using that Ci and by solving the question of the characteristic value of the equation (1). Then, the equation (1) is solved by calculating new F by using this Ci. This calculation is carried out repeatedly and is ended at last when there is almost no difference between the Ci used in the calculation of F and the found Ci. This method is called as SCF (self-consistent field) method and is widely used in the calculation of molecular orbits.
Because a number of two-electron integrals represented by the equation (1) is proportional to fourth power of the total number N of the atomic orbit, the value of N amounts around 1000 and the number of two-electron integrals amounts to the fourth power of that, i.e., in the order of 100 trillions, when a molecule composed of around 100 atoms which often appears in the field of biology for example. Although a method of judging and cutting off ones whose value is small before calculating the two-electron integrals has been often used here, a number of two-electron integrals which need to be calculated is around 100 millions which is still an enormous amount of number.
Therefore, because there is no memory space for calculating and storing the two-electron integrals once even though the same two-electron integrals are used in each repetition of the SCF method, a direct method of calculating the two-electron integrals again per each repetition is normally used. Because the most of the calculation time is occupied by the calculation of the two-electron integrals in the calculation of molecular orbit of this direct method, it is important to increase the speed of this part.
Here, a Gaussian function which allows the two-electron integrals to be found analytically is normally used for the atomic orbit xi represented by the equation (4). As a method for calculating the two-electron integrals at high speed using the atomic orbit of this Gaussian function, there has been known a method shown in a document 1, “S. Ohara and A. Sakai, J. Chem. Phys. 84, 3963 (1986)”(hereinafter referred to as Ohara's method).
The Ohara's method is represented by a format of recurrence formula containing auxiliary integration by introducing a value of the auxiliary integration expanding the two-electron integral. One two-electron integral may be expressed by the format of sum of products arithmetic operation containing lower order auxiliary integral by this recurrence formula. The two-electron integral may be found by developing into a format containing only the lowest order auxiliary integral in accordance to the recurrence formula and by finding the higher order auxiliary integral sequentially by the sum of products arithmetic operation. A concrete calculation method of the Ohara's method will be shown below.
At first, the atomic orbit x represented by the Gaussian function is represented by the following equation in the Ohara's method:
x
(
r−R; n
, &zgr;)=(
rx−Rx
)
nx
(
ry−Ry
)
ny
×(
rz−Rz
)
nz
exp[−&zgr;(
r−R
)
2
]  (6)
Here, r and R are vectors representing spatial positions. R in particular represents the center of atom. n is a vector composed of an integer greater than zero and is called as an orbital quantum number vector. This orbital quantum number vector has three components of nx, ny and nz of x, y and z similarly to r.
&zgr; is a constant cal

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