第四章單點計算與量化圖形

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第四章單點計算與量化圖形

所謂單點計算是指在 Born-Oppenheimer (通常 nonrelativistic) 假設下，指定所有原子核的位置與電子數後，求解電子運動的薛丁格方程式的計算，所得的能量包含電子的動能，位能，以及原子核的靜電排斥能量，但不包含原子核的動能。其相對的能量零點為所有原子核與電子互相分離到無窮遠且為靜止狀態。單點計算與量化圖形的產生過程大致如下：

(1) 建構 Input file (*.com, gjf)

提供分子結構，指定電子數目 (電荷)，計算理論，基底函數，其他計算選項及參數。

例如：

%NProcShared=4

%mem=64MW

%chk=BF3

#MP2/6-31+G(d,p) SCF=InCore Density=MP2 Test calculation using BF3

0 1 B

F 1 R1

F 1 R1 2 120.

F 1 R1 2 120. 3 180.

R1 = 1.3

接下來執行程式: (或由 G09W 執行) hu@Hubble NCHC]$ g09 BF3.com &

[1] + Done g09 BF3.com [hu@Hubble NCHC]$ ls

BF3.chk BF3.com BF3.log

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(2)檢視 Output file

Entering Gaussian System, Link 0=g09 Input=BF3.com

Output=BF3.log Initial command:

/home/hu/g09/l1.exe /s/hu/Gau-16489.inp -scrdir=/s/hu/

Entering Link 1 = /home/hu/g09/l1.exe PID= 16490.

This is part of the Gaussian(R) 09 program. It is based on the Gaussian(R) 03 system (copyright 2003, Gaussian, Inc.), the Gaussian(R) 98 system (copyright 1998, Gaussian, Inc.), the Gaussian(R) 94 system (copyright 1995, Gaussian, Inc.), the Gaussian 92(TM) system (copyright 1992, Gaussian, Inc.), the Gaussian 90(TM) system (copyright 1990, Gaussian, Inc.), the Gaussian 88(TM) system (copyright 1988, Gaussian, Inc.), the Gaussian 86(TM) system (copyright 1986, Carnegie Mellon University), and the Gaussian 82(TM) system (copyright 1983, Carnegie Mellon University). Gaussian is a federally registered trademark of Gaussian, Inc.

This software contains proprietary and confidential information, including trade secrets, belonging to Gaussian, Inc.

This software is provided under written license and may be used, copied, transmitted, or stored only in accord with that written license.

The following legend is applicable only to US Government contracts under FAR:

RESTRICTED RIGHTS LEGEND

Use, reproduction and disclosure by the US Government is subject to restrictions as set forth in subparagraphs (a) and (c) of the Commercial Computer Software - Restricted Rights clause in FAR 52.227-19.

Gaussian, Inc.

340 Quinnipiac St., Bldg. 40, Wallingford CT 06492

---

Warning -- This program may not be used in any manner that competes with the business of Gaussian, Inc. or will provide assistance to any competitor of Gaussian, Inc. The licensee of this program is prohibited from giving any competitor of Gaussian, Inc. access to this program. By using this program, the user acknowledges that Gaussian, Inc. is engaged in the business of creating and licensing software in the field of computational chemistry and represents and warrants to the licensee that it is not a competitor of Gaussian, Inc. and that

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it will not use this program in any manner prohibited above.

--- Cite this work as:

Gaussian 09, Revision A.02,

M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria,

M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr.,

J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand,

K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009.

******************************************

Gaussian 09: EM64T-G09RevA.02 11-Jun-2009 25-Jul-2014

******************************************

%NProcShared=4

Will use up to 4 processors via shared memory.

%mem=64MW %chk=BF3

---

#MP2/6-31+G(d,p) SCF=InCore Density=MP2 ---

1/30=1,38=1/1;

2/12=2,17=6,18=5,40=1/2;

3/5=1,6=6,7=111,11=9,16=1,25=1,30=1,71=1/1,2,3;

4//1;

5/5=3,38=5/2;

8/6=4,10=2/1;

9/15=1,16=-3/6;

10/5=1,13=10,20=2,30=1/2;

6/7=2,8=2,9=2,10=2,22=2/1;

99/5=1,9=1/99;

---

Test calculation using BF3 ---

Symbolic Z-matrix:

Charge = 0 Multiplicity = 1 B

F 1 R1

F 1 R1 2 120.

F 1 R1 2 120. 3 180. 0 Variables:

R1 1.3

Input orientation:

---

Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z

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---

1 5 0 0.000000 0.000000 0.000000 2 9 0 0.000000 0.000000 1.300000 3 9 0 1.125833 0.000000 -0.650000 4 9 0 -1.125833 0.000000 -0.650000 ---

Distance matrix (angstroms):

1 2 3 4 1 B 0.000000

2 F 1.300000 0.000000

3 F 1.300000 2.251666 0.000000

4 F 1.300000 2.251666 2.251666 0.000000 Stoichiometry BF3

Framework group D3H[O(B),3C2(F)]

Deg. of freedom 1

Full point group D3H NOp 12 Largest Abelian subgroup C2V NOp 4 Largest concise Abelian subgroup C2 NOp 2 Standard orientation:

---

Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z ---

1 5 0 0.000000 0.000000 0.000000 2 9 0 0.000000 1.300000 0.000000 3 9 0 1.125833 -0.650000 0.000000 4 9 0 -1.125833 -0.650000 0.000000 ---

Rotational constants (GHZ): 10.4935426 10.4935426 5.2467713 Standard basis: 6-31+G(d,p) (6D, 7F)

There are 34 symmetry adapted basis functions of A1 symmetry.

There are 7 symmetry adapted basis functions of A2 symmetry.

There are 22 symmetry adapted basis functions of B1 symmetry.

There are 13 symmetry adapted basis functions of B2 symmetry.

Integral buffers will be 131072 words long.

Raffenetti 1 integral format.

Two-electron integral symmetry is turned on.

76 basis functions, 128 primitive gaussians, 76 cartesian basis functions 16 alpha electrons 16 beta electrons

nuclear repulsion energy 112.0618691461 Hartrees.

NAtoms= 4 NActive= 4 NUniq= 2 SFac= 3.00D+00 NAtFMM= 80 NAOKFM=F Big=F

One-electron integrals computed using PRISM.

NBasis= 76 RedAO= T NBF= 34 7 22 13 NBsUse= 76 1.00D-06 NBFU= 34 7 22 13 ...

...

Initial guess orbital symmetries:

Occupied (E') (E') (A1') (A1') (A1') (E') (E') (A1') (E') (E') (A2") (E') (E') (E") (E") (A2')

Virtual (A2") (A1') (E') (E') (A2") (A1') (E') (E') (E") (E") (A1') (E') (E') (A2") (A1') (A2') (E') (E') (E') (E') (A2") (E') (E') (A1') (E") (E") (A1') (E") (E") (E') (E') (A2") (A1') (A2') (E') (E') (E') (E') (E") (E") (A1") (E') (E') (A1') (E') (E') (A2') (A1') (A2") (E") (E") (E') (E') (E') (E') (A1') (A1') (A1') (E') (E')

The electronic state of the initial guess is 1-A1'.

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Requested convergence on RMS density matrix=1.00D-08 within 128 cycles.

Requested convergence on MAX density matrix=1.00D-06.

Requested convergence on energy=1.00D-06.

No special actions if energy rises.

Keep R1 ints in memory in canonical form, NReq=5139001.

SCF Done: E(RHF) = -323.208916863 A.U. after 10 cycles Convg = 0.1643D-08 -V/T = 2.0026

ExpMin= 3.15D-02 ExpMax= 7.00D+03 ExpMxC= 1.05D+03 IAcc=3 IRadAn= 5 AccDes= 0.00D+00

HarFok: IExCor= 205 AccDes= 0.00D+00 IRadAn= 5 IDoV=-2 ScaDFX= 1.000000 1.000000 1.000000 1.000000

Range of M.O.s used for correlation: 5 76

NBasis= 76 NAE= 16 NBE= 16 NFC= 4 NFV= 0 NROrb= 72 NOA= 12 NOB= 12 NVA= 60 NVB= 60

**** Warning!!: The largest alpha MO coefficient is 0.11293407D+02 Fully in-core method, ICMem= 15658020.

JobTyp=1 Pass 1 fully in-core, NPsUse= 4.

Spin components of T(2) and E(2):

alpha-alpha T2 = 0.1981767387D-01 E2= -0.8384925841D-01 alpha-beta T2 = 0.9793787105D-01 E2= -0.4259786899D+00 beta-beta T2 = 0.1981767387D-01 E2= -0.8384925841D-01 ANorm= 0.1066570775D+01

E2 = -0.5936772067D+00 EUMP2 = -0.32380259406950D+03 ...

...

**********************************************************************

Population analysis using the MP2 density.

**********************************************************************

Orbital symmetries:

Occupied (E') (E') (A1') (A1') (A1') (E') (E') (A1') (E') (E') (A2") (E') (E') (E") (E") (A2')

Virtual (A1') (A2") (E') (E') (A2") (A1') (E') (E') (A1') (E") (E") (E') (E') (A2") (A2') (A1') (E') (E') (E') (E') (A2") (E') (E') (A1') (E") (E") (A1') (E') (E') (A1') (E") (E") (A2") (A2') (E') (E') (E') (E') (E") (E") (A1") (A1') (E') (E') (E') (E') (A2') (A1') (A2") (E") (E") (E') (E') (E') (E') (A1') (A1') (A1') (E') (E')

The electronic state is 1-A1'.

Alpha occ. eigenvalues -- -26.35738 -26.35738 -26.35736 -7.78765 -1.69774 Alpha occ. eigenvalues -- -1.65313 -1.65313 -0.86380 -0.82039 -0.82039 Alpha occ. eigenvalues -- -0.76978 -0.70503 -0.70503 -0.69072 -0.69072 Alpha occ. eigenvalues -- -0.66439

Alpha virt. eigenvalues -- 0.05156 0.06900 0.07978 0.07978 0.21533 Alpha virt. eigenvalues -- 0.29519 0.29730 0.29730 0.34297 0.34333 Alpha virt. eigenvalues -- 0.34333 0.36824 0.36824 0.37258 0.42644 Alpha virt. eigenvalues -- 0.48588 0.48683 0.48683 0.63490 0.63490 Alpha virt. eigenvalues -- 0.73378 0.93222 0.93222 1.08672 1.41390 Alpha virt. eigenvalues -- 1.41390 1.74087 1.76607 1.76607 1.78347 Alpha virt. eigenvalues -- 1.80467 1.80467 1.85395 1.89409 1.91239 Alpha virt. eigenvalues -- 1.91239 1.97667 1.97667 2.08690 2.08690 Alpha virt. eigenvalues -- 2.09410 2.14592 2.16038 2.16038 2.19248 Alpha virt. eigenvalues -- 2.19248 2.25030 2.32889 2.45451 2.81503

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Alpha virt. eigenvalues -- 2.81503 3.30818 3.30818 3.51205 3.51205 Alpha virt. eigenvalues -- 3.80449 4.16200 5.41168 5.54267 5.54267 Condensed to atoms (all electrons):

1 2 3 4

1 B 2.836395 0.303750 0.303750 0.303750 2 F 0.303750 9.278394 -0.082346 -0.082346 3 F 0.303750 -0.082346 9.278394 -0.082346 4 F 0.303750 -0.082346 -0.082346 9.278394 Mulliken atomic charges:

1

1 B 1.252356 2 F -0.417452 3 F -0.417452 4 F -0.417452

Sum of Mulliken atomic charges = 0.00000

Mulliken charges with hydrogens summed into heavy atoms:

1

1 B 1.252356 2 F -0.417452 3 F -0.417452 4 F -0.417452

Sum of Mulliken charges with hydrogens summed into heavy atoms = 0.00000 Electronic spatial extent (au): <R**2>= 209.6544

Charge= 0.0000 electrons

Dipole moment (field-independent basis, Debye):

X= 0.0000 Y= 0.0000 Z= 0.0000 Tot= 0.0000 Quadrupole moment (field-independent basis, Debye-Ang):

XX= -22.3600 YY= -22.3600 ZZ= -18.1022 XY= 0.0000 XZ= 0.0000 YZ= 0.0000 ...

...

N-N= 1.120618691461D+02 E-N=-9.945131237260D+02 KE= 3.223800929917D+02 Symmetry A1 KE= 2.037181594425D+02

Symmetry A2 KE= 6.288940396714D+00 Symmetry B1 KE= 1.009706361267D+02 Symmetry B2 KE= 1.140235702581D+01

1\1\GINC-HUBBLE\SP\RMP2-FC\6-31+G(d,p)\B1F3\HU\25-Jul-2014\0\\#MP2/6-3 1+G(d,p) SCF=InCore Density=MP2\\Test calculation using BF3\\0,1\B\F,1 ,1.3\F,1,1.3,2,120.\F,1,1.3,2,120.,3,180.,0\\Version=EM64T-G09RevA.02\

State=1-A1'\HF=-323.2089169\MP2=-323.8025941\RMSD=1.643e-09\Dipole=0., 0.,0.\Quadrupole=-1.0551951,2.1103902,-1.0551951,0.,0.,0.\PG=D03H [O(B 1),3C2(F1)]\\@

Sometimes the fool who rushes in gets the job done.

-- Al Bernstein

Job cpu time: 0 days 0 hours 0 minutes 10.5 seconds.

File lengths (MBytes): RWF= 5 Int= 0 D2E= 0 Chk= 1 Scr= 1 Normal termination of Gaussian 09 at Fri Jul 25 09:13:47 2014.

Atomic Units:

1 bohr =

a

₀

= ℏ

²

4 πε

₀

e

²

m

_e = 0.52918 Å

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1 hartree = ^h=

m_ee⁴ 4 ε02h²

1

n²= e²

4 πε₀a₀ = 627.5095 kcal/mol

GaussView 可直接讀取 Gaussian 09 之 output file，並顯示基本的運算結果。

若要檢視計算所得之 orbitals, density, electrostatic potential 等波函數相關資料，需先產生 formatted formchk 檔。 (GaussView 可直接讀取 G09W 之 checkpoint file)

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(3) 產生 Formatted checkpoint file (或者在 Route Section 加入 "FChk" keyword)

[hu@Hubble NCHC]$ formchk BF3.chk [hu@Hubble NCHC]$ ls

BF3.chk BF3.com BF3.fchk BF3.log

開啟 BF3.fchk, 由 Results --> Surface/Contour...

Electron Isodensity (0.0004) mapped with ESP

HOMO LUMO

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(4) 讀入非內建基底函數

雖然 Gaussian 有許多內建的基底函數，但有時候還是需要用到一些新的或 Gaussian 中沒有包含的 basis sets，最常見的情況是包含過渡金屬的計算，許多常用的基底函數不支援3d 以後的金屬，內建可支援重金屬的基底函數通常品質也並不理想。

以下是讀入 Peterson 等人於 2005 年發表的 aug-cc-pVDZ-pp basis set 計算金原子的範例：

#B3LYP Gen Pseudo=Read Au atom

0 2 Au

Au 0 S 7 1.00

38.0008000 0.0200090 23.9725000 -0.1516790 15.2182000 0.3639600 5.5399900 -0.8213260 1.3855100 0.9366410 0.6424610 0.4235270 0.1564960 0.0162500 S 7 1.00

38.0008000 -0.0053040 23.9725000 0.0463180 15.2182000 -0.1199400 5.5399900 0.3040620 1.3855100 -0.4945010 0.6424610 -0.2555160 0.1564960 0.6091630 S 7 1.00

38.0008000 0.0323400 23.9725000 -0.1717620 15.2182000 0.3395030 5.5399900 -0.7461790 1.3855100 1.9174980 0.6424610 -1.1299310 0.1564960 -1.3662840 S 1 1.00

0.0549100 1.0000000 S 1 1.00

0.0193000 1.0000000 P 6 1.00

10.3092000 0.1282170 6.6276500 -0.3537900 1.6744700 0.5662110 0.8011160 0.4931710 0.3468790 0.1137750 0.1227010 0.0025340 P 6 1.00

10.3092000 -0.0358850 6.6276500 0.1028900 1.6744700 -0.2009880 0.8011160 -0.2010450 0.3468790 0.1165490

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0.1227010 0.5949690 P 6 1.00

10.3092000 -0.0772420 6.6276500 0.2226110 1.6744700 -0.4953530 0.8011160 -0.3607470 0.3468790 0.7100000 0.1227010 0.5688390 P 1 1.00

0.0424280 1.0000000 D 5 1.00

11.0027000 0.0164670 6.8916600 -0.0680130 1.8080800 0.2994920 0.8210510 0.4543030 0.3441610 0.3442240 D 5 1.00

11.0027000 -0.0236280 6.8916600 0.0956720 1.8080800 -0.5412940 0.8210510 -0.3955330 0.3441610 0.6191650 D 1 1.00

0.1297430 1.0000000 D 1 1.00

0.0489000 1.0000000 F 1 1.00

0.8919000 1.0000000 F 1 1.00

0.3789000 1.0000000

****

AU 0

AU-ECP 5 60 h-ul potential 1

2 1.0000000 0.0000000 s-ul potential

2

2 13.5232180 426.6418670 2 6.2643840 36.8006680 p-ul potential

4

2 11.4138670 87.0020910 2 10.3292150 174.0043700 2 5.7074240 8.8706100 2 4.8281650 17.9024380 d-ul potential

4

2 7.4309630 49.8836550 2 8.3219900 74.6845490 2 4.6096420 6.4862270 2 3.5115070 9.5468210 f-ul potential

2

2 3.0846390 8.7916400 2 3.0247430 11.6584560

指令列中 "Gen"代表要求讀入 general basis set，Pseudo=Read 代表要輸入 Pseudo-

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Potential，因為在這個 basis set 中金原子的 60 個內層電子是由 Effective Core Potential (ECP) 所代表，只有外層 19 個電子以及 5s,5p,6s,5d,6p orbitals 是由基底函數來描述原子軌域。許多各式各樣的基底函數及 ECP 都可以由 EMSL 的網站

(https://bse.pnl.gov/bse/portal) 中找到並下載。

在計算中我們也可以混合基底函數，如下例Mo(CO)₆ 分子的計算:

%chk=MoCO6

%mem=800MW

%NprocShared=4

#B3LYP Gen Pseudo=Read SCF=InCore OPT FREQ Mo(CO)6

0 1

Mo 0.0 0.0 0.0 C 2.0 0.0 0.0 C -2.0 0.0 0.0 C 0.0 2.0 0.0 C 0.0 -2.0 0.0 C 0.0 0.0 2.0 C 0.0 0.0 -2.0 O 3.2 0.0 0.0 O -3.2 0.0 0.0 O 0.0 3.2 0.0 O 0.0 -3.2 0.0 O 0.0 0.0 3.2 O 0.0 0.0 -3.2 Mo 0

LanL2DZ

****

C,O 0 6-31G(d)

****

Mo 0 F 1 1.00

1.043 1.00

****

Mo 0 LANL2

對於 C, O 我們使用 6-31G(d) basis set，對 Mo 我們使用 LANL2DZ basis set，並讀入 LANL2 ECP，另外對於 Mo 我們加上額外一組 f polarization functions，以平衡此混合 的基底函數。通常不建議直接將較大型的基底函數如 6-311+G(d,p) 等直接與

LANL2DZ 結合，因為可能造成電子分布的不均衡，較建議直接使用 aug-cc-pVnZ-pp (n = D, T, Q) 基底函數

胡維平

國立中正大學

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化學暨生物化學系

第四章 單點計算與量化圖形

a

= ℏ

4 πε

e

m

第四章單點計算與量化圖形