# Electronic Structures - Chemistry

Electronic Structures - Chemistry

Our task: To learn how to do ab initio calculations by the Gaussian_98.

What means to do the quantum Chemical Calculations: - To get Total

energy and Total Wavefunction of the system

How we can obtain the E tot and WF: To solve the Schrodinger eq.

where:

H = E

H = T + N e - E

For many-electron systems, it is impossible exactly solve the Schrodinger eq. Therefore, some

approximations are necessary. The Hartree-Fock and Roothan approximations are among the

most popular approximations.

Hartree-Fock (HF) Approximation.- This is an approximation to the Hamiltonian H:

H ⇒ F = T + N e

+ V HF

Where V HF – is the average potential experienced by the i-th electron due to the presence of the

other electrons. The essence of the HF approximation is to replace the

complicated many-electron problem by a one-electron problem in which

electron-electron repulsion is treated in an average way.

As a result of HF approximation we are loosing electron correlation !!!

V HF depends from the wavefunction of i-th electron, so HF eq should be solved ONLY by Self-

Consistent-Field (SCF) technique.

Roothan approximation: This is approximation to the wavefunction (molecular orbitals):

χ µ

- is Atomic Orbitals (or called Basis sets).

ϕ I

= Σ C µι

χ µ

If we will combine HF and R-approximations, then we will get Hartree-Fock-Roothan eq.

FC = EC

So, we should know F and C (coefficient matrix) to get Total Energy and Wavefunction.

How to solve this eq. - Use SCF approach:

0. Choose the spin state, charge and geometry of the molecule

1. Choose basis set

2. Calculate every integrals, V HF using the chosen basis set

3. Calculate guess spin density matrix ρ

4. Solve HFR equation for this guess basis set and get energy and C 1

at the first

step.

5. Repeat (2) and (3)

6. Determine whether procedure has converged (for energy and density)

7. If NO ⇒ go to step 4;

2

If YES ⇒ go to 4 , get final Energy, C and F and finish.

Thus, choice of basis sets is VERY IMPORTANT ! Therefore, let me to give a

Ψ = |ϕ i

|

ϕ i

= Σ C iµ

χ µ

χ µ

- is a Atomic Orbitals (Basis sets).

In the mathematical sense, the best way to present of χ µ

is Slater Type of Functional (or

Orbitals) – STO – which centered at atomic centers

(1) Slater basis functions (STOs)

STO

~ e − r H − like

1s

STO ~ re − r

2 s

~

1s

H − like

2 s

H −like

STO

2 p x

~ xe − r =

2p x

STO

2 p y

~ ye − r =

2 p y

H −like

STO

2 p z

~ ze − r =

2p z

ζ = “exponent”

H −like

~ ( 2 − 2 r)e − r

However, practically it is very difficult to use STO basis sets for molecular calculations.

Therefore, people came to the idea to use the Gaussian Type of Functionals (Orbitals)- GTO.

Gaussian Basis Functions

• Formula of Cartesian GTO:

GTO

~ e − r2

s

GTO ~ xe − r2 , ye − r 2 , ze − r2

p

GTO ~ xy ⋅ e − r 2 , xz ⋅ e − r 2 , yz ⋅ e − r 2 , x 2 ⋅e − r2 , y 2 ⋅ e − r 2 , z 2 ⋅e − r 2 (6d)

d

or xy⋅ e − r 2 , xz ⋅ e − r 2 , yz ⋅e − r 2 , ( x 2 − y 2

)e − r 2 , ( 2z 2 − x 2 − y 2

)e − r 2 (5d)

6d functions contain extra ‘s’ like function.

f functions : (10f), (7f)

α = “exponent”

• Comparison between Gaussian-type orbitals and Slater-type orbitals

ξ and α are positive numbers, determining the diffuseness or “size” of function: large exp. ⇒

small functional (spatially)

1.

There are two major differences between STO and GTO:

• at r = 0

⎡ dx STO ⎤ ⎡ dx GTO ⎤

⎢ dr ⎥

< 0 ,

⎦ dr

r = 0

⎢ ⎥

= 0

r =0

“GTO does not satisfy the cusp condition at r=0.”

2. At r →∞ , GTO , exponential of -αr 2 decays much rapidly the STO , exponential of -αr

3

AO

Slater

Density

Gaussian

Slater

Gaussian

Thus, GTO’s are easier to use than STO, but GTO’s are NOT optimum basis functions,

because they have different functional behavior compared to the known functional behavior of

Atomic Orbitals (at r = 0 and r →∞). What we shall do People came to the idea to use several

GTO’s to better describe the known features of STO- functionals:

or

STO-NG basis sets:

STO ⇒ N GTO

AO

Slater

STO-3G

STO-2G

STO-1G

Another way is to use more Gaussian Functionals to as better as possible describe the

AO’s (people call this number of Gaussians as “primitive” functionals). Even if we use a large

number of (primitive) GTO’s, still we can reduce the computer time dramatically compared to

STO’s.

But computer time can be reduced, without sacrifing the accuracy of the calculations, by

combining primitive functions into several groups, called Contracted GTO’s (CGTO’s):

χ i CGF

= Σ d iµ

P µ

GTO

For example, we can take TWO GTO for describing S-orbital and combine them

together as:

χ 1s CGF = d 1

P 1

+ d 2

P 2

Since here we used TWO primitives to get χ 1s CGF , this type of basis set (orbital) called

“double-zeta” basis set, and so on.

4

Example in the case of carbon atom:

C 3 P : (1s) 2 (2s) 2 (2p) 2

Use 9 s-GTO’s describe 1s and 2s atomic orbitals

and 5 sets of p-GTO’s for 2p atomic orbitals :

9

9

1s

= ∑C , 1s ,r s ,r 2s

= ∑C 2 s,r s , r

r =1

5

r =1

(9s5p) GTOs

∑ p ,r

r =1

2 p xyz

= C 2 p ,r

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

HF solution

Contraction Coefficients

for C( 3 P) atom

of CGTO

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

α 1s 2s s 1

s 2

s 3

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

4233. 0.00122 0.00026 0.00122 0 0

634.9 0.00934 0.00202 0.00934 0 0

146.1 0.04534 0.0974 0.04545 0 0

42.50 0.15459 -0.03606 0.15466 0 0

14.19 0.35867 -0.08938 0.35887 0 0

5.148 0.43809 -0.17699 0.43887 -0.16837 0

1.967 0.14581 -0.05267 0.14592 0 0

0.4962 0.00199 0.57408 0 1.06009 0

0.1533 0.00041 0.54768 0 1.0

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

α 2p basis p 1

basis p 2

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

18.16 0.01469 0.01854 0

3.986 0.09150 0.011544 0

1.143 0.3011 0.38619 0

0.3594 0.50734 0.64011 0

0.1146 0.31735 0 1.0

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

Since basis sets can be interpreted as restriction of each electron to a particular region of

space, larger basis sets impose fewer constraints of electrons, and more accurately approximate

exact molecular orbital: Thus, accuracy of the calculations improves via increasing the basis sets

to HF limit:

Minimal Basis Set < Double-Zeta < Triple-Zeta < …………. HF basis limit

There are mixed approach too: To use minimal basis set for inner-shells, but double-,

triple-, … zeta basis sets for valence shells (electrons). This approach called – “split-valence”

basis sets: 3-21G, 6-31G etc.

Practically used CGTO basis sets

A. Minimal Basis Sets

Use 1 CGTO for each atomic shell

STO-NG

Fit one STO by N GTO’s

Actually STO-3G is used most popularly.

5

B. Double-Zeta and Valence Double-Zeta Basis Sets

Use 2 CGTOs for each atomic shell ⇒ DZ

However, use only 1 CGTOs for inner shell ⇒ VDZ or split valence

For second row atoms :

3-21G

3 means use one CGTO for inner-shell (core) orbital from three GTO’s.

21 means 3 GTO’s are contracted to 2 and 1 CGTO’s for valence orbitals (2s & 2p AOs).

i.e., (3s)/[1s] for core and (3s3p)/[2s2p] for valence

sss → (sss) for 1s AO

sss → (ss)(s) for 2s AO (split valence)

ppp → (pp)(p) for 2p AO (split valence)

6-31G

(6s)/[1s] for core, (4s4p)/[2s2p] for valence shell

more primitives = more accurate

same CGTO’S = VDZ

(slightly better than 3-21G)

ssssss → (ssssss) for 1s AO

ssss → (sss)(s) for 2s AO (split valence)

pppp → (ppp)(p) for 2p AO (split valence)

Double-zeta basis set

Huzinaga-Dunning DZ

–––––––––––––––––––––––––––––––––––––––––––––––––––

double zeta valence double

D95

D95V

–––––––––––––––––––––––––––––––––––––––––––––––––––

for 2nd row (9s5p)/[4s 2p] (9s 5p)/[3s 2p]

3rd row (11s7p)/[6s 4p] (11s7p)/[4s 3p]

for H atom (4s)/[2p] (4s)/[2p]

–––––––––––––––––––––––––––––––––––––––––––––––––––

• We need larger basis set with electron correlation.

6-311G

triple-zeta for valence shell orbitals

ssssss → (ssssss) for core orbital

sssss → (sss)(s)(s) for s-type valence orbitals

ppppp → (ppp)(p)(p) for p-type valence orbitals

6-31G** 6-31G(d,p)

first ‘*’ means adding d functions on non-H atoms.

second ‘*’ means adding p functions on H atoms.

polarization functions :

add p on H.

add d on C,N,O , Si, P, S. (Si, P, S atoms are more polarizable.)

add f for transition metals allowed to change forms (more flexible hybridization)

6

Mixing of polarization function is small, but gives more flexible wavefunction.

6-311G(3df,2p)

adding three d functions and one f functions on non-H atoms and

adding two p functions on H atoms.

6-311++G

first ‘+’ means adding diffuse function on non-H atom.

second ‘+’ means adding diffuse function on H atom.

Various options for Basis Set specification

The following basis sets are stored internally in the Gaussian 98 program.

♦ STO-3G

♦ 3-21G

♦ 6-21G

♦ 4-31G

♦ 6-31G

♦ 6-311G

♦ D95V : Dunning-Huzinaga valence double-zeta

♦ D95 : Dunning-Huzinaga full double zeta

♦ SHC : D95V on first row, Goddard/ Smedley ECP on second row

♦ CEP-4G : Stephens-Basch-Krauss ECP minimal basis

♦ CEP-31G : Stephens-Basch-Krauss ECP split valance

♦ CEP-121G : Stephens-Basch-Krauss ECP triple-split basis

♦ LanL2MB : STO - 3G on first row, Los Alamos ECP plus MBS on Na - Bi

♦ LanL2DZ : D95 on first row, Los Alamos ECP plus DZ on Na-Bi

♦ cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z : Dunning’s correlation consistent basis sets

(double, triple, quadruple, and quintuple - zeta, respectively). These basis sets include

polarization functions by definition. The following table lists the valence polarization

functions present for the various atoms included in these basis sets:

♦ aug-cc-pVDZ, aug-cc-pVTZ, aug-cc-pVQZ, aug-cc-pV5Z : Dunning’s correlation

consistent basis sets, augmented with diffuse functions.

Atoms cc-pVDZ cc-pVTZ cc-pVQZ cc-pV5Z

H 2s, 1p 3s, 2p, 1d 4s, 3p, 2d, 1f 5s, 4p, 3d, 2f, 1g

He 2s, 1p 3s, 2p, 1d 4s, 3p, 2d, 1f not available

B - He 3s, 2p, 1d 4s, 3p, 2d, 1f 5s, 4p, 3d, 2f, 1g 6s, 5p, 4d, 3f, 2g, 1h

Al - Ar 4s, 3p, 1d 5s, 4p, 2d, 1f 6s, 5p, 3d, 2f, 1g 7s, 6p, 4d, 3f, 2g, 1h

These basis sets may be augmented with diffuse functions by adding the aug- prefix

to the basis set keyword (rather than using the + and ++ notation)

Adding Polarization and Diffuse Functions

Single first polarization functions can also be requested using the usual * or ** notation. Note

that (d,p) and ** are synonymous – 6-31G** is equivalent to 6-31G(d,p), for example – and

that the 3-21G * basis set has polarization functions on second row atoms only. The + and ++

7

diffuse functions are available with some basis sets, as are multiple polarization functions. The

keyword syntax is best illustrated by example: 6-31+G(3df,2p) designates the 6-31G basis set

supplemented by diffuse functions, 3 sets of d functions and one set of f functions on heavy

atoms, and supplemented by 2 sets of p functions on hydrogens.

When the AUG- prefix is used to add diffuse function to the cc-pVxZ basis sets, one

diffuse function of each function type in use for a given atom is added. For example, the augcc-pVTZ

basis places one s, one d, and one p diffuse functions on hydrogen atoms, and one d,

one p, one d, and one f diffuse functions on B through Ne and AI through Ar.

Adding a single polarization function to 6-31G (i.e. 6-311G(d)) will result in one d function

for first and second row atoms and one function for first transition row atoms, since d functions

are already present for the valence electrons latter. Similarly, adding a diffuse function to the 6-

311G basis set will produce one s, one p, and one d diffuse functions for third - row atoms.

The following table lists polarization and diffuse function availability and the range of

applicability for each built-in basis set in Gaussian 94:

Polarization Diffuse

Basis set Available atoms Functions Functions

STO-3G H - Xe (d)

3-21G H - Xe (d) or (d, p) +

6-21G H - Cl (d)

4-31G H - Ne (d) or (d, p)

6-31G H - Cl (3df, 3pd) ++

6-311G H - Br (3df, 3pd) ++

D95 H - Cl (3df, 3pd) ++

D95V H - Cl (d) or (d, p) ++

SHC H - Cl (3df, 3pd) ++

CEP-4G H - Cl (3df, 3pd) ++

CEP-31G H - Cl (3df, 3pd) ++

CEP-121G H - Cl (3df, 3pd) ++

LanL2MB H - Ba, La - Bi

LanL2DZ H, Li - Ba, La - Bi

cc-pVxZ H - He, B - Ne, Al - Ar included in added via

8

Method Available in Gaussian_98

HF - Hartree-Fock

B3LYP - Density Functional, B3LYP, and more

DFT methods also available

MP2 - Moller-Plesset Second order

MP3 - Moller-Plesset Third order

MP4 (SDQ) - Moller-Plesset fourth order (and also

includes S, D and Q Excitations

MP4 (SDTQ) - Adds triple excitations

CISD - Configuration Interaction

QCISD - Quatratic CI

CCSD - Coupled-cluster method

CASSCF - Complete Active Space-SCF

Amber, UFF - Molecular Mechanics

ZINDO, MNDO,

AM1 and more, - Semi-empirical methods:

And more…

HF- limit

Basis Sets

Exact Solution

Basis Sets

Truncation

Error

Method (CI)

Truncation

Error

Full CI

Methods

9

chem430

Passwd: *******

UNIX

Tree Structure: Files and Directories

cd - change directory: cd directory_name

rm - remove command: rm file_name: also: rmdir, rm –I

mv - move comman: mv file_name

mkdir - make directory: mkdir directory_name

cp - copy command: cp file_name_1 file_name_2

ls - list command

VI

Command mode and Insert mode

vi file_name

Type i for insert command

Use backspace in order to correct mistake

:w :w! or :wq :wq! Write or write and quit

:q :q! quit

dd

n delete

o

Insert line

How to submit Jobs

1. Prepare your gaussian Input

2. Put the name of you input file into g98.cmd file

3. llsubmit g98.cmd

Water Input : (in /home/chemistry/ch_tch/chem430/h2o_inp)

# RHF/6-31G*

H20 SIngle Point Calculations

0 1

O

H 1 OH

H 1 OH 2 A1

OH 0.97

A1 105.0

10

Water Output :

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

Gaussian 98: IBM-RS6000-G98RevA.7 11-Apr-1999

9-Oct-2002

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

------------

# RHF/6-31G*

------------

1/38=1/1;

2/17=6,18=5/2;

3/5=1,6=6,7=1,11=1,25=1,30=1/1,2,3;

4/7=1/1;

5/5=2,32=1,38=4/2;

6/7=2,8=2,9=2,10=2,28=1/1;

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

-----------------------------

H20 SIngle Point Calculations

-----------------------------

Symbolic Z-matrix:

Charge = 0 Multiplicity = 1

O

H 1 OH

H 1 OH 2 A1

Variables:

OH 0.97

A1 105.

------------------------------------------------------------------------

Z-MATRIX (ANGSTROMS AND DEGREES)

CD Cent Atom N1 Length/X N2 Alpha/Y N3 Beta/Z J

------------------------------------------------------------------------

1 1 O

2 2 H 1 .970000( 1)

3 3 H 1 .970000( 2) 2 105.000( 3)

------------------------------------------------------------------------

Z-Matrix orientation:

---------------------------------------------------------------------

Center Atomic Atomic Coordinates (Angstroms)

Number Number Type X Y Z

---------------------------------------------------------------------

1 8 0 .000000 .000000 .000000

2 1 0 .000000 .000000 .970000

3 1 0 .936948 .000000 -.251054

---------------------------------------------------------------------

Distance matrix (angstroms):

1 2 3

1 O .000000

2 H .970000 .000000

3 H .970000 1.539105 .000000

Interatomic angles:

H2-O1-H3=105.

Stoichiometry H2O

Framework group C2V[C2(O),SGV(H2)]

Deg. of freedom 2

Full point group C2V NOp 4

Largest Abelian subgroup C2V NOp 4

11

Largest concise Abelian subgroup C2 NOp 2

Standard orientation:

---------------------------------------------------------------------

Center Atomic Atomic Coordinates (Angstroms)

Number Number Type X Y Z

---------------------------------------------------------------------

1 8 0 .000000 .000000 .118100

2 1 0 .000000 .769553 -.472399

3 1 0 .000000 -.769553 -.472399

---------------------------------------------------------------------

Rotational constants (GHZ): 809.6732555 423.3751244 278.0065412

Isotopes: O-16,H-1,H-1

Standard basis: 6-31G(d) (6D, 7F)

There are 10 symmetry adapted basis functions of A1 symmetry.

There are 1 symmetry adapted basis functions of A2 symmetry.

There are 3 symmetry adapted basis functions of B1 symmetry.

There are 5 symmetry adapted basis functions of B2 symmetry.

Crude estimate of integral set expansion from redundant integrals=1.400.

Integral buffers will be 262144 words long.

Raffenetti 1 integral format.

Two-electron integral symmetry is turned on.

19 basis functions 36 primitive gaussians

5 alpha electrons 5 beta electrons

nuclear repulsion energy 9.0725181920 Hartrees.

One-electron integrals computed using PRISM.

NBasis= 19 RedAO= T NBF= 10 1 3 5

NBsUse= 19 1.00D-04 NBFU= 10 1 3 5

Projected INDO Guess.

Initial guess orbital symmetries:

Occupied (A1) (A1) (B2) (A1) (B1)

Virtual (A1) (B2) (A1) (A1) (A1) (A1) (A1) (A1) (A2) (B1)

(B1) (B2) (B2) (B2)

Warning! Cutoffs for single-point calculations used.

Requested convergence on RMS density matrix=1.00D-04 within 64 cycles.

Requested convergence on MAX density matrix=1.00D-02.

Requested convergence on energy=5.00D-05.

Keep R1 integrals in memory in canonical form, NReq= 454104.

SCF Done: E(RHF) = -76.0097026179 A.U. after 6 cycles

Convg = .3112D-04 -V/T = 2.0032

S**2 = .0000

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

Population analysis using the SCF density.

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

Orbital Symmetries:

Occupied (A1) (A1) (B2) (A1) (B1)

Virtual (A1) (B2) (B2) (A1) (A1) (B1) (B2) (A1) (A1) (A2)

(B1) (A1) (B2) (A1)

The electronic state is 1-A1.

Alpha occ. eigenvalues -- -20.56258 -1.33487 -.70087 -.56760 -.49685

Alpha virt. eigenvalues -- .20802 .30071 1.01922 1.12526 1.15986

Alpha virt. eigenvalues -- 1.16796 1.37727 1.43141 2.02474 2.03648

Alpha virt. eigenvalues -- 2.07057 2.60687 2.92539 3.96479

Condensed to atoms (all electrons):

12

1 2 3

1 O 8.341554 .262244 .262244

2 H .262244 .322750 -.018015

3 H .262244 -.018015 .322750

Total atomic charges:

1

1 O -.866042

2 H .433021

3 H .433021

Sum of Mulliken charges= .00000

Atomic charges with hydrogens summed into heavy atoms:

1

1 O .000000

2 H .000000

3 H .000000

Sum of Mulliken charges= .00000

Electronic spatial extent (au): = 19.1078

Charge= .0000 electrons

Dipole moment (Debye):

X= .0000 Y= .0000 Z= -2.2270 Tot= 2.2270

XX= -7.2530 YY= -4.0828 ZZ= -5.9962

XY= .0000 XZ= .0000 YZ= .0000

Octapole moment (Debye-Ang**2):

XXX= .0000 YYY= .0000 ZZZ= -1.4282 XYY= .0000

XXY= .0000 XXZ= -.3752 XZZ= .0000 YZZ= .0000

YYZ= -1.4212 XYZ= .0000

XXXX= -5.2365 YYYY= -5.5383 ZZZZ= -6.1377 XXXY= .0000

XXXZ= .0000 YYYX= .0000 YYYZ= .0000 ZZZX= .0000

ZZZY= .0000 XXYY= -2.0762 XXZZ= -1.9476 YYZZ= -1.5987

XXYZ= .0000 YYXZ= .0000 ZZXY= .0000

N-N= 9.072518192003D+00 E-N=-1.986097369779D+02 KE= 7.577092287728D+01

Symmetry A1 KE= 6.772251853541D+01

Symmetry A2 KE= 2.287797476747D-34

Symmetry B1 KE= 4.556826042745D+00

Symmetry B2 KE= 3.491578299122D+00

1\1\GINC-PP14\SP\RHF\6-31G(d)\H2O1\CHEM430\09-Oct-2002\0\\# RHF/6-31G*

\\H20 SIngle Point Calculations\\0,1\O\H,1,0.97\H,1,0.97,2,105.\\Versi

on=IBM-RS6000-G98RevA.7\State=1-A1\HF=-76.0097026\RMSD=3.112e-05\Dipol

e=0.6951155,0.,0.5333809\PG=C02V [C2(O1),SGV(H2)]\\@

By Jamal Musaev

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