原子虛位勢的Ab Initio計算

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原子虛位勢的 Ab Initio 計算

Ab Initio Computation for Atomic Pseudopotential

研 究 生：林浙于

指導教授：葉立明 教授

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原子虛位勢的 Ab Initio 計算

Ab Initio Computation for Atomic Pseudopotential

研 究 生：林浙于 Student：Che-Yu Lin 指導教授：葉立明 Advisor：Li-Ming Yeh 國 立 交 通 大 學 應 用 數 學 系 碩 士 論 文 A Thesis

Submitted to Department of Applied Mathematics College of Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master

in

Applied Mathematics July 2009

Hsinchu, Taiwan, Republic of China

原子虛位勢的 Ab Initio 計算

學生：林浙于 指導教授：葉立明 國立交通大學應用數學系﹙研究所﹚碩士班 摘要 我們用 ab initio self-consistent 方法計算出每一個原子的基態電子密度,然後再利用基態電子 密度得出原子的基態總能.我們利用基態電子密度為基礎,構造出原子中每個電子組態的波函數 所相對應的虛波函數,最後求得虛波函數的虛位勢.

Ab Initio Computation for Atomic Pseudopotential

student：Che-Yu Lin Advisors：Dr. Li-Ming Yeh

Department of Applied Mathematics

National Chiao Tung University

ABSTRACT

We calculate the ground-state electronic density of each atom by using ab initio self-consistent method, and then we obtain the total ground state energy of atoms from using the ground-state electronic density of each atom. We can use the ground-state electronic density of each atom to construct the pseudo wave function which corresponds to the wave function of valence states. In the end, we can obtain the pseudopotential of the pseudo wave function for each angular momentum.

誌

謝

非常感謝我的指導教授,對於我研究上的問題,不厭其煩的為我解答,陪我一起成長,無以回 報,並且謝謝各位口試委員的指導,還要感謝陪我一起奮鬥的同學,在遇到困難或灰心的時候,給 我鼓勵和支持,我的家人也是對我不遺餘力的支持,如果沒有這些幫助我的人,我也不能順利的 完成學業,對此,我感恩於心。

Contents

Abstract (in Chinese) i

Abstract (in English) ii

Acknowledgement iii

Contents iv

Tables vi

Figures vii

1. Preface 1

2. The premise of physics 2

2.1 The Schrödinger equation...2

2.2 Density functional theory and the Kohn-Sham equation...4

2.3 The local density approximation...5

2.4 Pseudopotential... ...6

3. Process of study 8

3.1 The expression of the potential of the Kohn-Sham equation...8

3.2 The simplification of the effective potential...9

3.3 The simplification of the Kohn-Sham equation...11

3.4 The discretization of the Kohn-Sham equation ...12

3.6 The total ground state energy and simplification...15

3.7 Construction of pseudopotential...18

A Appendix 26

A.1 The contents of the main program of No atom……….26

A.2 The initial electron density of No atom(1*401)………...61

A.3 The diagrams of output resulting………..65

Tables

Table 1 Comparison of E[n]………....17 Table 2 The choices of the cutoff radius rc for each states of z=1 to z=25

atoms...19 Table 3 The choices of the cutoff radius rc for each states of z=26 to z=50

atoms……….………….……..…20 Table 4 The choices of the cutoff radius rc for each states of z=51 to z=75

atoms………....21 Table 5 The choices of the cutoff radius rc for each states of z=76 to z=102

Figures

Figure 1 The computational procedure for the calculation of the ground state

density………14

Figure 2 The ratio of

₍

[ ] HF

₎

HF tot tot E n −E E ………...………16

Figure 3 The real and pseudo wave functions of H………...66

Figure 4 The real and pseudo wave functions of He………...…….66

Figure 5-6 The real and pseudo wave functions of Li………..66

Figure 7-8 The real and pseudo wave functions of Be………...…..67

Figure 9-11 The real and pseudo wave functions of B………...67

Figure 12-14 The real and pseudo wave functions of C………...67

Figure 15-17 The real and pseudo wave functions of N………..…68

Figure 18-20 The real and pseudo wave functions of O……….………68

Figure 21-23 The real and pseudo wave functions of F………..……69

Figure 24-26 The real and pseudo wave functions of Ne………..…………..69

Figure 27-30 The real and pseudo wave functions of Na………..…………..70

Figure 31-34 The real and pseudo wave functions of Mg………..….70

Figure 35-39 The real and pseudo wave functions of Al………...70

Figure 40-44 The real and pseudo wave functions of Si………..…71

Figure 50-54 The real and pseudo wave functions of S………...72

Figure 55-59 The real and pseudo wave functions of Cl……….…73

Figure 60-63 The real and pseudo wave functions of Ar………...……..73

Figure 64-69 The real and pseudo wave functions of K………..………75

Figure 70-75 The real and pseudo wave functions of Ca……….75

Figure 76-82 The real and pseudo wave functions of Sc………..…...76

Figure 83-89 The real and pseudo wave functions of Ti…………..…………77

Figure 90-96 The real and pseudo wave functions of V………..………78

Figure 97-103 The real and pseudo wave functions of Cr………..………….79

Figure 104-110 The real and pseudo wave functions of Mn…………..……..80

Figure 111-117 The real and pseudo wave functions of Fe………...………...80

Figure 118-124 The real and pseudo wave functions of Co………...…..81

Figure 125-131 The real and pseudo wave functions of Ni……….82

Figure 132-137 The real and pseudo wave functions of Cu………....83

Figure 138-144 The real and pseudo wave functions of Zn………..…..84

Figure 145-152 The real and pseudo wave functions of Ga………..…..85

Figure 153-160 The real and pseudo wave functions of Ge………...86

Figure 161-168 The real and pseudo wave functions of As………...…..87

Figure 177-184 The real and pseudo wave functions of Br………...89

Figure 185-192 The real and pseudo wave functions of Kr………...…..90

Figure 193-201 The real and pseudo wave functions of Rb……….91

Figure 202-210 The real and pseudo wave functions of Sr………..…92

Figure 211-220 The real and pseudo wave functions of Y………...…93

Figure 221-230 The real and pseudo wave functions of Zr………...…...94

Figure 231-240 The real and pseudo wave functions of Nb…………..……..95

Figure 241-250 The real and pseudo wave functions of Mo………..…..97

Figure 251-260 The real and pseudo wave functions of Tc…………..……...98

Figure 261-270 The real and pseudo wave functions of Ru…………..……..99

Figure 271-280 The real and pseudo wave functions of Rh…………..……100

Figure 281-290 The real and pseudo wave functions of Pd………...…102

Figure 291-300 The real and pseudo wave functions of Ag………...…103

Figure 301-310 The real and pseudo wave functions of Cd………...104

Figure 311-321 The real and pseudo wave functions of In………..…..105

Figure 322-332 The real and pseudo wave functions of Sn…………..…….107

Figure 333-343 The real and pseudo wave functions of Sb…………..…….108

Figure 344-355 The real and pseudo wave functions of Te………...……….110

Figure 368-378 The real and pseudo wave functions of Xe………….…..…113

Figure 379-390 The real and pseudo wave functions of Cs………..……….114

Figure 391-402 The real and pseudo wave functions of Ba………..….115

Figure 403-415 The real and pseudo wave functions of La………...117

Figure 416-428 The real and pseudo wave functions of Ce………..….119

Figure 429-442 The real and pseudo wave functions of Pr………....120

Figure 443-456 The real and pseudo wave functions of Nd………..…122

Figure 457-470 The real and pseudo wave functions of Pm…………..……124

Figure 471-484 The real and pseudo wave functions of Sm………..………125

Figure 485-497 The real and pseudo wave functions of Eu………...…….127

Figure 498-512 The real and pseudo wave functions of Gd………..………129

Figure 513-526 The real and pseudo wave functions of Tb…………..…….131

Figure 527-540 The real and pseudo wave functions of Dy………..……....132

Figure 541-555 The real and pseudo wave functions of Ho………..…134

Figure 556-569 The real and pseudo wave functions of Er…………..…….136

Figure 570-583 The real and pseudo wave functions of Tm………...138

Figure 584-597 The real and pseudo wave functions of Yb………...140

Figure 598-611 The real and pseudo wave functions of Lu………..….141

Figure 626-639 The real and pseudo wave functions of Ta………..….145

Figure 640-651 The real and pseudo wave functions of W…………..…….147

Figure 652-665 The real and pseudo wave functions of Re…………..……148

Figure 666-679 The real and pseudo wave functions of Os………..…150

Figure 680-693 The real and pseudo wave functions of Ir………..…..152

Figure 694-707 The real and pseudo wave functions of Pt………..…..153

Figure 708-721 The real and pseudo wave functions of Au………...155

Figure 722-734 The real and pseudo wave functions of Hg…………..……157

Figure 735-749 The real and pseudo wave functions of Ti…………..……..158

Figure 750-765 The real and pseudo wave functions of Pb……..………….160

Figure 766-780 The real and pseudo wave functions of Bi……...………….162

Figure 781-795 The real and pseudo wave functions of Po………..……….164

Figure 796-810 The real and pseudo wave functions of At………..……….166

Figure 811-823 The real and pseudo wave functions of Rn………..…168

Figure 824-839 The real and pseudo wave functions of Fr………..….170

Figure 840-856 The real and pseudo wave functions of Ra………..…172

Figure 857-873 The real and pseudo wave functions of Ac…………..……174

Figure 874-890 The real and pseudo wave functions of Th…………..……176

Figure 909-926 The real and pseudo wave functions of U…………..……..180 Figure 927-944 The real and pseudo wave functions of Np………..………183 Figure 945-962 The real and pseudo wave functions of Pu………..….185 Figure 963-980 The real and pseudo wave functions of Am………...…...…187 Figure 981-997 The real and pseudo wave functions of Cm………...……189 Figure 998-1015 The real and pseudo wave functions of Bk………...……..192 Figure 1016-1033 The real and pseudo wave functions of Cf…………..….194 Figure 1034-1051 The real and pseudo wave functions of Es………..…….196 Figure 1052-1069 The real and pseudo wave functions of Fm………..……198 Figure 1070-1087 The real and pseudo wave functions of Md………….….201 Figure 1088-1105 The real and pseudo wave functions of No………...203