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Development of efficient algorithms for quantum chemistry calculations of large molecules
Development of efficient algorithms for quantum chemistry calculations of large molecules
石村, 和也
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イシムラ, カズヤ
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ISHIMURA, Kazuya
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総合研究大学院大学
博士（理学）
Quantum chemistry plays an important role in elucidating molecular geometries,<br /> electronic states, and reaction mechanisms, because of the developments of a variety of<br /> theoretical methods, such as Hartree-Fock (HF), Møler-Plesset (MP) perturbation,<br /> configuration interaction (CI), coupled-cluster (CC), and density functional theory <br /> (DFT) methods. Electronic structure calculations have been carried out by not only<br /> theoretical chemists but also experimental chemists. DFT is currently most widely used<br />to investigate large molecules in the ground state as well as small molecules because of the low computational cost. However, the generally used functionals fail to describe<br />correctly non-covalent interactions that are important for host-guest molecules,<br /> self-assembly, and molecular recognition, and they tend to underestimate reaction<br /> barriers. Many attempts have been made to develop new functionals and add<br /> semiempirical or empirical correction terms to standard functionals, but no generally<br /> accepted DFT method has emerged yet.<br /> Second-order Møler-Plesset perturbation theory (MP2) is the simplest method that<br /> includes electron correlation important for non-covalent interactions and reaction<br /> barriers nonempirically. However, the computational cost of MP2 is considerably<br /> higher than that of DFT. In addition, much larger sizes of fast memory and hard disk<br /> are required in MP2 calculations. These make MP2 calculations increasingly difficult<br /> for larger molecules. Since workstation or personal computer (PC) clusters have<br /> become popular for quantum chemistry calculations, an efficient parallel calculation is<br /> a solution of the problem. Therefore, new parallel algorithms for MP2 energy and<br /> gradient calculations are presented in this thesis. Furthermore, an efficient algorithm<br /> for the generation of two-electron repulsion integrals (ERIs) which is important in <br /> quantum chemistry calculations is also presented.<br /> For the calculations of excited states, different approaches are required: for<br /> example, CI, multi-configuration self-consistent field (MCSCF), time-dependent DFT<br /> (TDDFT), and symmetry adapted cluster (SAC)/SAC-CI methods. One of the most<br /> accurate methods is SAC/SAC-CI, as demonstrated for many molecules. In this thesis,<br /> SAC/SAC-CI calculations of ground, ionized, and excited states are presented.<br /> This thesis consists of five chapters: a new algorithm of two-electron repulsion<br /> integral calculations (Chapter I), a new parallel algorithm of MP2 energy calculations<br /> (Chapter II), a new parallel algorithm of MP2 energy gradient calculations (Chapter<br /> III), applications of MP2 calculations (Chapter IV), and SAC/SAC-CI calculations of <br /> ionized and excited states (Chapter V).<br /> In quantum chemistry calculations, the generation of ERIs is one of the most basic<br /> subjects and is the most time-consuming step especially in direct SCF calculations.<br /> Many algorithms have been developed to reduce the computational cost. In<br /> Pople-Hehre algorithm, Cartesian axes are rotated to make several coordinate<br /> components zero or constant, so that these components are skipped in the generation of ERIs. In McMurchie-Davidson algorithm, ERIs are generated from (<i>ss</i>|<i>ss</i>) type<br /> integrals using a recurrence relation derived from Hermite polynomials. By combining<br /> these two algorithms, a new algorithm is developed in Chapter I. The results show that<br /> the new algorithm reduces the computational cost by 10 - 40%, as compared with the<br /> original algorithms. It is notable that the generation of ERIs including d functions is<br /> considerably fast. The program implemented officially in GAMESS in 2004 has been<br /> used all over the world.<br /> In quantum mechanics, perturbation methods can be used for adding corrections<br /> to reference solutions. In the MP perturbation method, a sum over Fock operators is<br /> used as the reference term, and the exact two-electron repulsion operator minus twice<br /> the average two-electron repulsion operator is used as the perturbation term. It is the<br /> advantage that the MP perturbation method is size consistent and size extensive, unlike<br /> truncated CI methods. The zero-order wave function is the HF Slater determinant, and <br /> the zero-order energy is expressed as a sum of occupied molecular orbital (MO)<br /> energies. The first-order perturbation is the correction for the overcounting of<br /> two-electron repulsions at zero-order, and the first-order energy corresponds to the HF<br /> energy. The MP correlation starts at second-order. In general, second-order (MP2) <br /> accounts for 80 - 90% of electron correlation. Therefore, MP2 is focused in this thesis<br /> since it is applicable to large molecules with considerable reliability and low <br /> computational cost.<br /> The formal computational scaling of MP2 energy calculations with respect to<br /> molecular size is fifth order, much higher than that of DFT energy calculations.<br /> Therefore, less expensive methods, such as Local MP2, density fitting (resolution of<br /> identity, RI) MP2, and Laplace Transform MP2, have been developed. However, all of<br /> these methods include approximations or cut-offs that need to be checked against full<br /> MP2 energies. An alternative approach to reduce the computational cost is to<br /> parallelize MP2 energy calculations. A number of papers on parallel MP2 energy<br /> calculations have been published. Almost all of them are based on simple <br /> parallelization methods that distribute only atomic orbital (AO) or MO indices to each <br /> processor. These methods have a disadvantage since intermediate integrals are <br /> broadcasted to all CPUs or the same AO ERIs are generated in all processors. Baker <br /> and Pulay developed a new parallel algorithm using SaebøAlmlöf integral<br /> transformation method. This algorithm parallelizes the first half transformation by AO<br /> indices and the second half transformation by MO indices. The advantages are that the<br /> total amount of network communication is independent of the number of processors <br /> and the AO integrals are generated only once. The disadvantage is the I/O overhead for<br /> the sorting of half-transformed integrals. A new parallel algorithm for MP2 energy<br /> calculations based on the two-step parallelization idea is presented in Chapter II. In<br /> this algorithm, AO indices are distributed in the AO integral generation and the first<br /> three quarter transformation, and MO indices are distributed in the last quarter<br /> transformation and MP2 energy calculation. Because the algorithm makes the sorting<br /> of intermediate integrals very simple, the parallel efficiency is highly improved and <br /> the I/O overhead is removed. Furthermore, the algorithm reduces highly the floating<br /> point operation (FLOP) count as well as the required memory and hard disk space, in <br /> comparison with other algorithms. Test calculations of taxol (C<small>47</small>H<small>51</small>NO<small>14</small>) and <br /> luciferin (C<small>11</small>H<small>8</small>N<small>2</small>O<small>3</small>S<small>2</small>) were performed on a cluster of Pentium 4 computers<br /> connected by gigabit Ethernet. The parallel scaling of the developed code is excellent<br /> up to the largest number of processors we have tested. For instance, the elapsed time<br /> for the MP2 energy calculations on 16 processors is on average 15.4 times faster than<br /> that on the single-processor.<br /> Determination of molecular geometries and reaction paths is a fundamental task in<br /> quantum chemistry and requires energy gradients with respect to nuclear coordinates. <br /> In Chapter III, a new parallel algorithm for MP2 energy gradient calculations is <br /> presented. The algorithm consists of 5 steps, the integral transformation, the MP2<br /> amplitude calculation, the MP2 Lagrangian calculation, the coupled-perturbed HF <br /> calculation, and the integral derivative calculation. All steps are parallelized by <br /> distributing AO or MO indices. The algorithm also reduces the FLOP count, the<br /> required memory, and hard disk space. Test calculations of MP2 energy gradients were <br /> performed for taxol and luciferin on a cluster of Pentium 4 computers. The speedups <br /> are very good up to 80 CPU cores we have tested. For instance, the speedup ratios are <br /> 28.2 - 33.0 on 32 processors, corresponding to 88% - 103% of linear speedup. This <br /> indicates the high parallel efficiency of the present algorithm. The calculation of taxol<br /> with 6-31G(d) (1032 contracted basis functions) finishes within 2 hours on 32<br /> processors, which requires only 1.8GB memory and 13.4GB hard disk per processor.<br /> Therefore, geometry optimization of molecules with 1000 basis functions can be easily<br /> performed using standard PC clusters.<br /> In Chapter IV, several applications of MP2 are performed using the program<br /> developed in Chapters II and III. Some molecules that DFT cannot treat well are <br /> optimized at the MP2 level. Geometry optimization is also carried out using the<br /> spin-component scaled (SCS) MP2 method. In this method, a different scaling is<br /> employed for the same and opposite spin components of the MP2 energy, so that<br /> SCS-MP2 performs as well as the much more costly CCSD(T) method at a high level<br /> of theory.<br />SAC theory is developed for ground states and based on CC theory that describes<br /> higher-order electron correlation. The main factor of electron correlation is collisions<br /> of two electrons. In CC theory, most collisions of four electrons can be taken in as the<br /> product of collisions of two electrons. Only a symmetry adapted excitation operator is <br /> used for the SAC expansion. Since the operator of the SAC expansion is totally<br /> symmetric, the unlinked terms (the products of the operators) are also totally <br /> symmetric. SAC-CI is developed to treat excited states. SAC and SAC-CI wave <br /> functions are orthogonal and Hamiltonian-orthogonal to each other. These<br /> orthogonalities are especially important for the calculations of transitions<br /> and relaxations. In general, the SAC-CI operators <i>R</i> are restricted to single and double<br /> excitations. This is called the SAC-CI SD-R method. For the calculations of high-spin<br /> states and multiple excitation processes, triple, quadruple, and higher excitation<br /> operators are included. This is called the SAC-CI general-R method. In Chapter V, the<br /> ground, singlet and triplet excited, ionized and electron attached states of ferrocene <br /> (Fe(C<small>5</small>H<small>5</small>)<small>2</small>) were calculated using the SAC/SAC-CI SD-R method. The calculated<br /> results are in good agreement with experimental values. It is found that shake-up<br /> processes (one electron ionization and one electron excitation) contribute to the first<br /> two ionization peaks.
総研大乙第178号
thesis
2007-09-28
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https://ir.soken.ac.jp/records/249
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