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If each molecular orbital is expanded by the basis functions and expressed as,

The Hartree-Fock equations are expressed as,

where

is a two electron integral. The suffixes

The density matrix element is expressed as

and contains the variations

In the classical programs such as Gaussian 70 used the two electron integral file with the packed four suffixes we utilize the file iteratively during the SCF calculations [1, 2]. From the symmetries of the suffixes, there are integrals of the same value. These are eliminated from the calculations and therefore there are six types of contributions to the Fock matrix element from a two electron integral as following:

Raffenetti has proposed a more efficient procedure to calculate the two-electron contribution to the Fock matrix, nowadays widely known as P super matrix algorithm [3]. Hereafter we call it P method, and another traditional four suffixes method as NOP method. The basis of the P method is to make a recombination of two electron integrals like,

The contribution to the Fock matrix element using P method becomes very simple.

If the number of two electron integrals does not change before and after the recombination and if the overhead for the recombination is small enough, the computational time will be much faster. The total number of multiply/add instructions would have been greatly reduced because the instructions decrease from six to two. The P method did work well, and both Hondo [2] and Gaussian [4] series of programs incorporate P method. Nowadays, the direct SCF method [5], which does not store the two electron integrals but calculates them repeatedly, is usually used. The P method, therefore, does not study well if the method works in any case.

On the other hand, due to the recent development of the microprocessor, it becomes possible to utilize the personal computer cluster to make the Hartree-Fock molecular orbital calculations with the parallel processing of the two electron integrals and the Fock matrix mentioned above. Because we now can use many CPUs and large size of memories that could not be supposed previously, it sometimes happens to break a previous common sense, that is, a

We are recently developing the molecular dynamics calculations based on the

Table 1. The configuration of PC cluster.

CPU | 8CPU Pentium4 2.4GHz, 512K Cache |

Chipset | Intel 845 |

Memory | 1 GB DDR266/ board |

Network | 1000 BaseT Ethernet |

Hard Disk | 60GB / 5400rpm |

Operating System | Linux kernel 2.4/RedHat 8.0 |

Fortran Compiler | Gnu Fortran77 |

Parallel Library | MPICH ver.1.2.4 |

The computational speed is measured with a series of minor-tranquilizers with the benzodiazepin and thienodiazepin frameworks; flutoprazepam (**1**, C_{19}H_{16}ClFN_{2}O), triazolam (**2**, C_{17}H_{12}Cl_{2}N_{4}), clothiazepam (**3**, C_{16}H_{15}ClN_{2}OS), etizolam (**4**, C_{17}H_{16}ClN_{4}S), flutazolam (**5**, C_{19}H_{18}ClFN_{2}O_{3}), and lorazepam (**6**, C_{15}H_{10}Cl_{2}N_{2}O_{2}) molecules (Scheme 1). The 3-21G basis set [8] is used throughout this study. We repeat the calculations ten times of single SCF and gradient of each molecule and take the fastest time among them.

Molecule | Flutoprazepam | Toriazolan | Clotiazepam | Etizolam | Fultazolam | Lorazepam |
---|---|---|---|---|---|---|

formula | C_{19}H_{16}ClFN_{2}O | C_{17}H_{12}C_{l2}N_{4} | C_{16}H_{15}ClN_{2}OS | C_{17}H_{16}ClN_{4}S | C_{19}H_{18}ClFN_{2}O_{3} | C_{15}H_{10}Cl_{2}N_{2}O_{2} |

atoms^{a} | 40 | 35 | 36 | 38 | 44 | 31 |

Basis^{b} | 248 | 239 | 227 | 245 | 274 | 217 |

TEI^{c} | 1.5GB | 1.3GB | 1.0GB | 1.2GB | 2.9GB | 0.9GB |

Table 3 shows the CPU and wall clock time for the P and NOP methods, respectively. In all cases for *N*=1, it is clearly shown that the wall clock time by the NOP method is shorter and 0.35-0.65 times smaller than that of the P method. Furthermore, when comparing the sum of CPU and system time, the NOP method shows shorter time, except for the results of fultazolam that are almost the same.

Table 3. Lists of CPU, system and wall clock time of P and NOP methods. N is number of CPUs.

Molecule | NOP | P | NOP/P Ratio | ||||||
---|---|---|---|---|---|---|---|---|---|

N | CPU | SYS | Wall | CPU | SYS | Wall | CPU+SYS | Wall | |

clotiazepam | 1 | 124.98 | 21.23 | 259.14 | 113.34 | 51.18 | 750.53 | 0.89 | 0.35 |

2 | 64.69 | 8.63 | 94.58 | 59.71 | 26.06 | 320.55 | 0.85 | 0.30 | |

4 | 34.38 | 4.62 | 50.58 | 32.04 | 10.64 | 100.82 | 0.91 | 0.50 | |

8 | 19.20 | 2.72 | 32.50 | 18.10 | 4.84 | 33.74 | 0.96 | 0.96 | |

etizolam | 1 | 157.29 | 28.89 | 397.69 | 142.25 | 72.07 | 994.16 | 0.87 | 0.40 |

2 | 81.44 | 11.91 | 119.48 | 76.82 | 36.00 | 444.19 | 0.83 | 0.27 | |

4 | 43.16 | 5.98 | 62.57 | 41.40 | 16.23 | 163.19 | 0.85 | 0.38 | |

8 | 23.92 | 3.63 | 38.68 | 23.22 | 6.59 | 42.18 | 0.92 | 0.92 | |

flutazolam | 1 | 265.37 | 88.77 | 1078.06 | 233.41 | 116.05 | 1660.53 | 1.01 | 0.65 |

2 | 145.96 | 39.85 | 451.09 | 124.32 | 56.85 | 795.94 | 1.03 | 0.57 | |

4 | 73.22 | 15.66 | 113.64 | 66.61 | 28.44 | 340.35 | 0.94 | 0.33 | |

8 | 40.35 | 8.98 | 66.96 | 37.30 | 12.13 | 113.36 | 1.00 | 0.59 | |

flutoprazepam | 1 | 179.72 | 38.16 | 482.96 | 163.63 | 82.20 | 1134.93 | 0.89 | 0.43 |

2 | 92.95 | 12.83 | 136.09 | 87.73 | 41.13 | 521.62 | 0.82 | 0.26 | |

4 | 49.53 | 6.92 | 71.23 | 46.18 | 17.12 | 197.99 | 0.89 | 0.36 | |

8 | 27.65 | 4.15 | 41.48 | 26.30 | 7.10 | 46.97 | 0.95 | 0.88 | |

lorazepam | 1 | 109.40 | 18.92 | 220.66 | 96.37 | 43.55 | 564.10 | 0.92 | 0.39 |

2 | 56.41 | 8.37 | 84.96 | 50.68 | 18.63 | 198.83 | 0.93 | 0.43 | |

4 | 31.59 | 4.68 | 46.70 | 26.62 | 7.92 | 45.65 | 1.05 | 1.02 | |

8 | 17.32 | 2.56 | 28.92 | 14.70 | 4.57 | 29.17 | 1.03 | 0.99 | |

triazolam | 1 | 152.96 | 29.02 | 401.02 | 138.81 | 67.93 | 963.88 | 0.88 | 0.42 |

2 | 78.96 | 11.62 | 114.63 | 71.59 | 33.13 | 431.96 | 0.86 | 0.27 | |

4 | 44.16 | 5.98 | 63.46 | 39.22 | 15.03 | 152.92 | 0.92 | 0.41 | |

8 | 24.09 | 3.68 | 37.05 | 21.51 | 6.38 | 39.88 | 1.00 | 0.93 |

Concerning the wall clock time, in all molecules, the difference becomes smaller when the number of CPUs increases. After all two electron integrals are buffered on the main memory, the wall clock time by the P method decreases more quickly than that by the NOP method showing the difference between the two methods. In the case of lorazepam, that is the smallest calculation of the present work, the difference between two methods disappears when 4 CPUs are used. In other molecules, the difference between the two methods also disappears when 8 CPUs are used again except for the fultazolam case. In the case of fultazolam, the NOP method result is still 0.59 times shorter than that of the P method even in the 8 CPU case, and the difference does not disappear in the present study. However, we consider from Table 3 that the difference between the two methods will vanish as the number of CPUs increases.

Table 4. The numbers of two electron integrals for NOP and P method.

Molecule | NOP | P | NOP/P Ratio |
---|---|---|---|

clotiazepam | 91696888 | 176425145 | 0.52 |

etizolam | 114762339 | 227059480 | 0.51 |

flutazolam | 192258830 | 366232404 | 0.52 |

flutoprazepam | 130469433 | 255888062 | 0.51 |

lorazepam | 82657615 | 154668492 | 0.53 |

triazolam | 114937886 | 219850389 | 0.52 |

The difference in the wall clock times between the two methods is brought by the difference of the amount of the files of the two electron integrals. We usually handle just the two electron integral that is larger than a certain threshold value (10^{-8} in the present study). Table 4 shows the number of two electron integrals by the P and NOP methods used in the present calculations. It is worthwhile to note that the number by the P method is almost two times larger than that by NOP method. From the definition of the P method, *I _{rstu}* has a certain value if the integral <

NOP | P | NOP/P ratio | |||||||
---|---|---|---|---|---|---|---|---|---|

Basis Set | N | CPU | SYS | Wall | CPU | SYS | Wall | CPU+SYS | Wall |

6-31G* | 1 | 17.68 | 3.15 | 26.39 | 11.90 | 3.76 | 19.90 | 1.33 | 1.33 |

2 | 9.12 | 1.70 | 16.10 | 6.32 | 1.98 | 11.20 | 1.30 | 1.44 | |

4 | 4.83 | 0.98 | 9.08 | 3.46 | 1.01 | 8.68 | 1.30 | 1.05 | |

8 | 2.86 | 0.66 | 6.66 | 2.02 | 0.65 | 5.32 | 1.32 | 1.25 | |

Number of Integrals | 20868299 | 23940759 | Ratio | 0.87 | |||||

3-21G | 1 | 1.52 | 0.36 | 3.14 | 1.07 | 0.46 | 2.75 | 1.23 | 1.14 |

2 | 0.85 | 0.28 | 2.53 | 0.64 | 0.27 | 2.33 | 1.24 | 1.09 | |

4 | 0.52 | 0.19 | 2.48 | 0.42 | 0.23 | 2.37 | 1.09 | 1.05 | |

8 | 0.39 | 0.17 | 2.91 | 0.33 | 0.18 | 2.89 | 1.10 | 1.01 | |

Number of Integrals | 2124399 | 277210 | Ratio | 0.76 |

In the present paper, we have studied the CPU time and the wall clock time required for the *ab initio* Hartree-Fock molecular orbital calculations with and without the Raffenetti's P super matrix algorithm under the parallel environment using the PC cluster. As realistic examples, the six different drug molecules of the minor-tranquilizer and the 3-21G basis set are used. In almost all of the cases, the P method cannot calculate faster than the NOP method in such a large calculation. It should be concluded that the P method sometimes calculates faster but sometimes does not. In large scale of calculations, it should be suggested to perform a test calculation to confirm which method is faster prior to the real calculations.

We are grateful to Dr. Kazumasa Shinjo and Dr. Shinsuke Shimogawa of ATR Adaptive Communication Laboratories for stimulating discussions and suggestions. This work was partially supported by Telecommunication Advancement Organization of Japan (TAO).

W. J, Hehre, L. Radom, P. v. R, Schleyer, and J. A. Pople,

[ 2] M. Dupuis, J. Rys, and H. F. King,

[ 3] R. C. Raffenetti,

[ 4] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, Jr., R. E. Stratmann, J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G. A. Petersson, P. Y. Ayala, Q. Cui, K. Morokuma, P. Salvador, J. J. Dannenberg, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V. Ortiz, A. G. Baboul, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, J. L. Andres, C. Gonzalez, M. Head-Gordon, E. S. Replogle, and J. A. Pople,

[ 5] J. Almlf, J. K. Faegri, and K. Korsell,

[ 6] M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. J. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. Su, T. L. Windus, M. Dupuis, J. A. Montgomery,

[ 7] H. Teramae and K. Ohtawara,

[ 8] J. S. Binkley, J. A. Pople, W. J. Hehre,

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