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 manyelectron systems, it is impossible exactly solve the Schrodinger eq. Therefore, some
approximations are necessary. The HartreeFock and Roothan approximations are among the
most popular approximations.
HartreeFock (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 ith electron due to the presence of the
other electrons. The essence of the HF approximation is to replace the
complicated manyelectron problem by a oneelectron problem in which
electronelectron 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 ith electron, so HF eq should be solved ONLY by Self
ConsistentField (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 Rapproximations, then we will get HartreeFockRoothan 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
little bit more information about the basis sets
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 Gaussiantype orbitals and Slatertype 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
Radius
Radius
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
STONG basis sets:
STO ⇒ N GTO
AO
Slater
STO3G
STO2G
STO1G
Radius
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 Sorbital 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
“doublezeta” basis set, and so on.
4
Example in the case of carbon atom:
C 3 P : (1s) 2 (2s) 2 (2p) 2
Use 9 sGTO’s describe 1s and 2s atomic orbitals
and 5 sets of pGTO’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 < DoubleZeta < TripleZeta < …………. HF basis limit
There are mixed approach too: To use minimal basis set for innershells, but double,
triple, … zeta basis sets for valence shells (electrons). This approach called – “splitvalence”
basis sets: 321G, 631G etc.
Practically used CGTO basis sets
A. Minimal Basis Sets
Use 1 CGTO for each atomic shell
STONG
Fit one STO by N GTO’s
Actually STO3G is used most popularly.
5
B. DoubleZeta and Valence DoubleZeta 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 :
321G
3 means use one CGTO for innershell (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)
631G
(6s)/[1s] for core, (4s4p)/[2s2p] for valence shell
more primitives = more accurate
same CGTO’S = VDZ
(slightly better than 321G)
ssssss → (ssssss) for 1s AO
ssss → (sss)(s) for 2s AO (split valence)
pppp → (ppp)(p) for 2p AO (split valence)
Doublezeta basis set
HuzinagaDunning 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.
6311G
triplezeta for valence shell orbitals
ssssss → (ssssss) for core orbital
sssss → (sss)(s)(s) for stype valence orbitals
ppppp → (ppp)(p)(p) for ptype valence orbitals
Adding polarization function
631G** 631G(d,p)
first ‘*’ means adding d functions on nonH 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.
6311G(3df,2p)
adding three d functions and one f functions on nonH atoms and
adding two p functions on H atoms.
Adding diffuse function
6311++G
first ‘+’ means adding diffuse function on nonH 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.
♦ STO3G
♦ 321G
♦ 621G
♦ 431G
♦ 631G
♦ 6311G
♦ D95V : DunningHuzinaga valence doublezeta
♦ D95 : DunningHuzinaga full double zeta
♦ SHC : D95V on first row, Goddard/ Smedley ECP on second row
♦ CEP4G : StephensBaschKrauss ECP minimal basis
♦ CEP31G : StephensBaschKrauss ECP split valance
♦ CEP121G : StephensBaschKrauss ECP triplesplit 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 NaBi
♦ ccpVDZ, ccpVTZ, ccpVQZ, ccpV5Z : 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:
♦ augccpVDZ, augccpVTZ, augccpVQZ, augccpV5Z : Dunning’s correlation
consistent basis sets, augmented with diffuse functions.
Atoms ccpVDZ ccpVTZ ccpVQZ ccpV5Z
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 – 631G** is equivalent to 631G(d,p), for example – and
that the 321G * 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: 631+G(3df,2p) designates the 631G 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 ccpVxZ basis sets, one
diffuse function of each function type in use for a given atom is added. For example, the augccpVTZ
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 631G (i.e. 6311G(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 builtin basis set in Gaussian 94:
Polarization Diffuse
Basis set Available atoms Functions Functions
STO3G H  Xe (d)
321G H  Xe (d) or (d, p) +
621G H  Cl (d)
431G H  Ne (d) or (d, p)
631G H  Cl (3df, 3pd) ++
6311G H  Br (3df, 3pd) ++
D95 H  Cl (3df, 3pd) ++
D95V H  Cl (d) or (d, p) ++
SHC H  Cl (3df, 3pd) ++
CEP4G H  Cl (3df, 3pd) ++
CEP31G H  Cl (3df, 3pd) ++
CEP121G H  Cl (3df, 3pd) ++
LanL2MB H  Ba, La  Bi
LanL2DZ H, Li  Ba, La  Bi
ccpVxZ H  He, B  Ne, Al  Ar included in added via
8
Method Available in Gaussian_98
HF  HartreeFock
B3LYP  Density Functional, B3LYP, and more
DFT methods also available
MP2  MollerPlesset Second order
MP3  MollerPlesset Third order
MP4 (SDQ)  MollerPlesset fourth order (and also
includes S, D and Q Excitations
MP4 (SDTQ)  Adds triple excitations
CISD  Configuration Interaction
QCISD  Quatratic CI
CCSD  Coupledcluster method
CASSCF  Complete Active SpaceSCF
Amber, UFF  Molecular Mechanics
ZINDO, MNDO,
AM1 and more,  Semiempirical methods:
And more…
HF limit
Basis Sets
Exact Solution
Basis Sets
Truncation
Error
Method (CI)
Truncation
Error
Full CI
Methods
9
Login:
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/631G*
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: IBMRS6000G98RevA.7 11Apr1999
9Oct2002
**********************************************

# RHF/631G*

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 Zmatrix:
Charge = 0 Multiplicity = 1
O
H 1 OH
H 1 OH 2 A1
Variables:
OH 0.97
A1 105.

ZMATRIX (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)

ZMatrix 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:
H2O1H3=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: O16,H1,H1
Standard basis: 631G(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.
Twoelectron integral symmetry is turned on.
19 basis functions 36 primitive gaussians
5 alpha electrons 5 beta electrons
nuclear repulsion energy 9.0725181920 Hartrees.
Oneelectron integrals computed using PRISM.
NBasis= 19 RedAO= T NBF= 10 1 3 5
NBsUse= 19 1.00D04 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 singlepoint calculations used.
Requested convergence on RMS density matrix=1.00D04 within 64 cycles.
Requested convergence on MAX density matrix=1.00D02.
Requested convergence on energy=5.00D05.
Keep R1 integrals in memory in canonical form, NReq= 454104.
SCF Done: E(RHF) = 76.0097026179 A.U. after 6 cycles
Convg = .3112D04 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 1A1.
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
Quadrupole moment (DebyeAng):
XX= 7.2530 YY= 4.0828 ZZ= 5.9962
XY= .0000 XZ= .0000 YZ= .0000
Octapole moment (DebyeAng**2):
XXX= .0000 YYY= .0000 ZZZ= 1.4282 XYY= .0000
XXY= .0000 XXZ= .3752 XZZ= .0000 YZZ= .0000
YYZ= 1.4212 XYZ= .0000
Hexadecapole moment (DebyeAng**3):
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
NN= 9.072518192003D+00 EN=1.986097369779D+02 KE= 7.577092287728D+01
Symmetry A1 KE= 6.772251853541D+01
Symmetry A2 KE= 2.287797476747D34
Symmetry B1 KE= 4.556826042745D+00
Symmetry B2 KE= 3.491578299122D+00
1\1\GINCPP14\SP\RHF\631G(d)\H2O1\CHEM430\09Oct2002\0\\# RHF/631G*
\\H20 SIngle Point Calculations\\0,1\O\H,1,0.97\H,1,0.97,2,105.\\Versi
on=IBMRS6000G98RevA.7\State=1A1\HF=76.0097026\RMSD=3.112e05\Dipol
e=0.6951155,0.,0.5333809\PG=C02V [C2(O1),SGV(H2)]\\@
By Jamal Musaev