Quantum course

Quantum Chemistry
Quantum Chemistry is the application of quantum mechanics to solve problems in
chemistry. It has been applied in different branches of chemistry
Physical Chemistry: To calculate thermodynamic properties, interpretation of molecular
spectra and molecular properties (e.g. Bond length, bond angles, …..etc.).
Organic Chemistry: To estimate the relative stabilities of molecules, and reaction
mechanism.
Analytical Chemistry: To Interpret of the frequency and intensity of line spectra.
Inorganic Chemistry: To predict and explain of the properties of transition metal
complexes.

Physical Chemistry
Inorganic Chemistry
Organic Chemistry
Photochemistry
Polymer
Surface and Catalysis
Drug Design
Toxicity

Historical background of quantum mechanics :
Nature of light:
Hertz, 1888, has showed that light is electromagnetic waves.
λ=c/ν
where, λ is the wavelength, c is the speed of light =2.998 x 1010 cm/sec, ν is the
frequency cm/sec.
Max Plank has assumed only certain quantities of light energy (E) could be emitted.
E = hν
Where, h is Plank’s constant = 6.6 x 10-27 erg.sec. The energy is quantized.

Photoelectric effect : Light comes out by shining surface in vacuum.
In 1905, Einstein, light can exhibit particle like behavior, called photons.
Ephoton = hν
hν = W + ½ mv2
where, W is the work function (minimum energy required to take electron out). ½
mv2 is the kinetic energy of emitted electron. From above, it is assumed that the

{Light looks like a particle and a wave}

Nature of Matter:
Rutherford and Geiger have found that some α-particles bounced right back
from golden foil, have small positive nucleus in atom.
In 1913, Bohr has studied the H-atom and assumed that the energy of electron
is quantized,
ν = ∆E / h
where, ν is the frequency of absorbed or emitted light, ∆E is the energy
difference between two states.
In 1923, DeBroglie has suggested that the motion of electrons might have a
wave aspect.
λ = h / mv = h / p
where, m is mass of electron. p is a particle momentum.
Accordingly, it has been suggested that electrons behave in some respect like
particles and in some others like waves. This is what is called a

Particle –Wave Duality.
The question arises, how can an electron be a particle, which is a localized
entity, and a wave, which is nonlocalized?
The answer is No, neither a wave nor a particle but it is something else.
The Classical physics has failed to describe the microscopic particles.

The question arises, how can an electron be a particle,
which is a localized entity and a wave which is a non-localized?

The answer is

No, neither a wave nor a particle but it is something else.

Heisenberg Uncertainty Principle:
"It is impossible to determine precisely and simultaneously the
momentum and the position of an electron
The statistical definition for the uncertainties is:

∆x . ∆px ≥ ħ / 2
where, = ħ / 2π

∆x . ∆px ≥ h / 4π

Werner Heisenberg
Nobel prize 1932

Wave Function:
To describe the state in quantum mechanics, we postulate the existence of a function of the
coordinates called the wave function (State function),

ψ.

ψ = ψ (x, t)
It contains all information about a system. The probability of finding a particle in a given place
can be given by ψ (Probability description).

What ψ (x) means?
-

ψ is an amplitude, sometimes complex function, not measurable, imaginary value.

-

ψψ* is a complex function, which may be real, and positive.

-

ψ has no physical meaning but ψψ* is the probability of locating the electron at a
given position.

If the probability of a certainty is defined as unity, this means:

If we have two different wave functions, ψ1 and ψ2 will be Normalized function when:
and

But if

or

The function is called orthogonal function.
But if
Where,

(called Kronecker Delta) is equal zero when i ≠ j and equal one when i = j, the

function is called orthonormalized function.

Time-dependent Schrödinger equation:
we postulate the existence of a function of the coordinates called the wave function
(State function), ψ. For one particle, one-dimensional system:
ψ= ψ (x, t)
It contains all information about a system. The probability of finding a particle in a
given place can be given by ψ (Probability

description).

Born postulates | ψ (x,t) |2 dx is the probability of finding a particle at position x and
at time t ( Probability density ).
ψ must satisfy Schrödinger equation. As t passes, ψ changes to differential
equation:

−  ∂Ψ ( x, t ) −  2 ∂ 2 Ψ ( x, t )
=
+ V ( x , t ) Ψ ( x, t )
2
i
∂t
2m
∂x
where, i=
, m = particle mass, V(x,t) = potential energy.
This is called Time-dependent Schrödinger equation.

Erwin Schrödinger
Nobel prize 1933

Schrödinger equation can be solved by the technique called separation
of variables:
the partial derivatives of this equation:

Making the substitution in equation 2:

Dividing by

Taking the left side of equation (3):
On integration:

ln

C is a constant of integration

One of the properties of the wave function, it is a complex, i.e.
where,

is a complex conjugate of

The complex conjugate of a function is the same function with a different sign of
imaginary value.

for stationary state

is called the Probability Density ( Time-independent wave function).

By equating the right side of equation (3) to a constant E, we have:

Time-independent Schrödinger
equation for a single particle of mass m moving in one

dimension. The constant E has the dimension of energy. In fact, it is
postulated that E is the energy of the system.

Operators:
Basis of quantum mechanics set up around two things:
1- Wave function, which contains all information about the system.
2- Operators which are rules whereby given some function, we can find another.

This operator is called the Hamiltonian operator for the system.
Kinetic energy =
Ĥ = Kinetic energy + Potential energy
=

So, the Eigen value equation:

Ĥ ψ(x) = E ψ(x)

-A particle in one-dimensional box:

I

V=0

III

II
X =0

Ψ=0
X =1

X

−  2 ∂ 2ψ ( x)
+ ∞ψ ( x) = Eψ ( x)
2
2m ∂x
∂ 2ψ ( x) 2m
+ 2 ( E − ∞)ψ ( x) = 0
∂x 2


We conclude that ψ(x) is zero outside the box:
ψ I(x) = zero

ψ III(x) = zero

For region II (inside the box), x between zero and l, the potential energy V(x) is zero,
and the Schrödinger equation becomes

n= 1,2,3,………

Fig. 3.1. The wave functions for
the 0ne-dimensional particle-ina-box

Fig. 3.2. The probability densities in 0nedimensional particle-in-a-box

II- The Harmonic Oscillator:
1- Try to understanding of molecular vibrations, their spectra and their
influence on thermodynamic properties.
2- Providing a good demonstration of mathematical techniques that are
important in quantum chemistry.
V

E

Velocity=0

-a

0

a

x

Fig. 4 The Parabolic Potential Energy of the Harmonic Oscillator. The classically
allowed (|x| ≤ a) and forbidden (|x| > a) regions for the Harmonic Oscillator

The classical force F is:
F= -kx
Where, F is a restoring force, k is a force constant, and x is a displacement on
x-axis.
F=

By integration:

Where,

= - kx

V(x) = ½ kx2

The Schrödinger equation

ψ(x) = E ψ(x), after multiplication on by

1

 α  4 −αx 2 2
ψ0 =   ∈
π 

The energy of a harmonic oscillator is quantized.

En = (n + ½ ) h ν

where n= 0,1,2…

III- The Hydrogen atom:
Ignoring interatomic or intermolecular interactions,
The isolated hydrogen atom is a two-particle system.
Instead of treating just the hydrogen atom,
we consider a slight more general problem,
the hydrogen-like atom. An exact solution of the Schrödinger equation
for atoms with more than one electron cannot be obtained because of
the interelectronic repulsions.

V = -Z é 2 / r

Where, V is the potential energy, Zé is the charge of nucleus,
(For Z=1, we have the hydrogen atom, for Z=2 the He + ion, for Z=3, the Li+ ion, etc…).
é is the proton charge in statocoulombs or as:
é≡
where, e is the proton charge in coulomb. To deal with the internal motion of the system,
we introduce µ as the mass of the particle.
µ = m e mn / m e + m n
where, me and mn are the electronic and nuclear masses.

where,

is Laplacian operator:

So, the time-independent Schrödinger equation is:

z

me

z = r cos θ
x = r sin θ . cos φ
y = r sin θ . sin φ

mn

x

θ r
φ

y

To solve this equation, we have to know that this wave is a spherical one, so, we should
convert the Cartesian coordinates to spherical polar coordinates.

There are two different variables in Schrödinger equation, one is the radial variable
(r) and the other is the angular variable
.

This is called Bohr radius. According to the Bohr theory, it is the radius of the circle in
which the electron moved in the ground state of the hydrogen atom.

The wave function for the ground state of the H-atom, where n=1, l=0, and m=0

The bound-state energy levels of the hydrogen-like atom are given by

Substituting the values of the physical constants into the energy equation of H-atom,
we find for (Z=1) ground state energy: E = -13.598 eV
(eV= electron volt)

Shapes of electron cloud:

Probability densities for some hydrogen-atom states

The overlap integral between two wave functions can be represented as S ij

Sij = ∫ψi ψj dτ Three different kinds of overlap are shown in Fig. ( 9).
Positive (Bonding)

Fig. 9 Three different kinds of overlap between two wave functions, ψ i and ψ j

Molecular Orbital Theory

:

The MO Theory has five basic rules:
1-The number of molecular orbitals = the number of atomic orbitals combined
of the two MO's, one is a bonding orbital (lower energy) and one is an anti-bonding orbital
(higher energy)
2-Electrons enter the lowest orbital available
3-The maximum # of electrons in an orbital is 2 (Pauli Exclusion Principle)
4- Electrons spread out before pairing up (Hund's Rule)

Heteronuclear molecules:
Hydrogen Fluoride:

Z

F
Y

Table 5

1σ

2σ

3π

4π

5σ

2s

-0.93

0.47

0

0

0.55

2px

-0.009

-0.68

0

0

0.80

2py

0

0

1.0

0

0

2pz

0

0

0

1.0

0

1sH

-0.16

-0.57

0

0

-1.05

E(eV)

-40.17

-15.39

-12.64

-12.64

3.20

H

X

2 0 0 0
NN
0.000000
.000000
.000000
2 1.9237-20.330
2 1.9170-14.540
0 0.0000 00.000
.0000 .00000.0000
7 30 2 0 3
1.020000
0.000000
0.000000
2 1.9237-20.330
2 1.9170-14.540
0 0.0000 00.000
.0000 .00000.0000
7 30 2 0 3
0 2 1
.010000
3.000000

Quantum chemical studies of the correlation diagram of N2 molecule with standard parameters:
__________________________________________________________________________________
ATOM

X

Y

Z

S
P
D
CONTRACTED D
N EXP COUL
N EXP COUL
N EXPD1 COUL C1
C2 EXPD2 AT EN S P D
N 1 .00000 .00000 .00000 2 1.9237 -20.3300 2 1.9170 -14.5400 0 .0000 .0000 .00000 .00000 .0000 7 30 2 0 3
N 2 1.20000 .00000 .00000 2 1.9237 -20.3300 2 1.9170 -14.5400 0 .0000 .0000 .00000 .00000 .0000 7 30 2 0 3
0CHARGE= 0 IEXIT= 0 ITYPE= 0 IPUNCH= 0 CON= 2.250 AU= .52920 SENSE= 2.000 MAXCYC= 25 DELTAC=.00000
0DISTANCE MATRIX
1
2
1 .0000 1.200
2 1.200 .0000
0TWO BODY REPULSION ENERGY MATRIX
1
2
1 .0000 4.4351
2 4.4351 .0000
4.43514361
0SPIN= 0

ENERGY LEVELS (EV)
E( 1) = 15.24199 0
E( 2) = -7.40075 0
E( 3) = -7.40075 0
E( 4) = -14.54000 2
E( 5) = -14.54000 2
E( 6) = -16.54000 2
E( 7) = -18.54000 2
E( 8) = -27.51826 2
0
ENERGY= -187.27312236 EV.
5 ORBITALS FILLED
0 HALF FILLED
0WAVE FUNCTIONS
0MO'S IN COLUMNS, AO'S IN ROWS
1
2
3
4
5
6
7
8

1
2
3
.0000 .0000 .0000
-1.2989 .0000 .0000
.0000 -.7443 .0000
.0000 .0000 .7443
.0000 .0000 .0000
-1.2989 .0000 .0000
.0000 .7443 .0000
.0000 .0000 -.7443

4
.0000
.0000
.0000
.5234
.0000
.0000
.0000
.5234

5
.0000
.0000
.5234
.0000
.0000
.0000
.5234
.0000

6
7
8
.0000 .5068 .5223
.5144 .0000 .0000
.0000 .0000 .0000
.0000 .0000 .0000
.0000 -.5068 .5223
-.5144 .0000 .0000
.0000 .0000 .0000
.0000 .0000 .0000

0REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM
1
2
1 4.0437 2.7000
2 2.7000 4.0437
0REDUCED CHARGE MATRIX, MO'S IN COLUMNS, ATOMS IN

ROWS

1
2
3
4
5
6
7
8
1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0ATOM
N 1
N 2

NET CHG.
.00000
.00000

S

ATOMIC ORBITAL OCCUPATION FOR GIVEN MO OCCUPATION
X
Y
Z
X2-Y2 Z2
XY
XZ
YZ

1.40947 1.59053 1.00000 1.00000
1.40947 1.59053 1.00000 1.00000

Quantum chemical studies of the correlation diagram of N2 molecule with the hybridized parameters:
__________________________________________________________________________________
ATOM

X

Y

Z

S
N EXP COUL
N 1 .00000 .00000 .00000 2 1.9237 -20.3300
N 2 1.02000 .00000 .00000 2 1.9237 -20.3300
0CHARGE= 0 IEXIT= 0 ITYPE= 0 IPUNCH= 0 CON=
0DISTANCE MATRIX
1
2
1 .0000 1.0200
2 1.0200 .0000
1
2
1 .0000 4.4351
2 4.4351 .0000
4.43514361
0SPIN=

0

P
D
CONTRACTED D
N EXP COUL N EXPD1 COUL C1 C2 EXPD2 AT EN S P D
2 1.9170 -14.5400 0 .0000 .0000 .00000 .00000 .0000 7 30 2 0 3
2 1.9170 -14.5400 0 .0000 .0000 .00000 .00000 .0000 7 30 2 0 3
2.250 AU= .52920 SENSE= 2.000 MAXCYC= 25 DELTAC=.00000

ENERGY LEVELS (EV)

E( 1) = 115.24199 0
E( 2) = -7.40075 0
E( 3) = -7.40075 0
E( 4) = -14.07184 2
E( 5) = -18.08657 2
E( 6) = -18.08873 2
E( 7) = -18.08873 2

E( 8) = -27.51826 2
0
ENERGY= -187.27312236 EV.
5 ORBITALS FILLED
0 HALF FILLED
0WAVE FUNCTIONS
1
2
3
4
5
6
7
8

1
2
3
-1.6157 .0000 .0000
-1.2989 .0000 .0000
.0000 -.7443 -.4460
.0000 -.4460 .7443
1.6157 .0000 .0000
-1.2989 .0000 .0000
.0000 .7443 -.4460
.0000 .4460 -.7443

4
5
.3190 -.4437
-.6477 .5144
.0000 .0000
.0000 .0000
.3190 .4437
.6477 -.5144
.0000 .0000
.0000 .0000

6
.0000
.0000
.3426
.5068
.0000
.0000
.3426
.5068

7
8
.0000 .5223
.0000 .1333
.5068 .0000
-.3426 .0000
.0000 .5223
.0000 -.1333
.5068 .0000
-.3426 .0000

REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM
1
2
1 4.0437 1.9126
2 1.9126 4.0437

ROWS

1
2
3
4
5
6
7
8
1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0ATOM
N 1
N 2

NET CHG.
.00000
.00000

S

X
Y
Z
X2-Y2 Z2
XY
XZ
YZ

1.40947 1.59053 1.00000 1.00000
1.40947 1.59053 1.00000 1.00000

3 0 0 0
OHH
2 2.2459-28.480
2 2.2266-13.620
0 0.0000 0.000
0.00000.00000.0000
8 35 2 4 0
1 1.2000-13.600
0 0.0000 0.000
0 0.0000 0.000
0.00000.00000.0000
1 10 1 0 0
1 1.2000-13.600
0 0.0000 0.000
0 0.0000 0.000
0.00000.00000.0000
1 10 1 0 0
0.00000000 0.00000000
1 2 0.99000
52.25000
1 3 0.99000
52.25000
0 1 2 2 1.000 132.0000

0.00000000
0.00000
180.00000

Quantum chemical studies of the electronic structure of H2O molecule:
__________________________________________________________________________________
IEXIT = 0
ETA .00000
ETA 180.00000
ATOM
X
Y
Z
S
P
D
CONTRACTED D
N EXP COUL N EXP COUL N EXPD1 COUL C1 C2 EXPD2 AT EN S P D
O 1 .00000 .00000 .00000 2 2.2459 -28.4800 2 2.2266 -13.6200 0 .0000 .0000 .00000 .00000 .0000 8 35 2 4 0
H 2 .78278 .00000 -.60610 1 1.2000 -13.6000 0 .0000 .0000 0 .0000 .0000 .00000 .00000 .0000 1 10 1 0 0
H 3 -.78278 .00000 -.60610 1 1.2000 -13.6000 0 .0000 .0000 0 .0000 .0000 .00000 .00000 .0000 1 10 1 0 0
0CHARGE= 0 IEXIT= 0 ITYPE= 0 IPUNCH= 0 CON= 2.250 AU= .52920 SENSE= 2.000 MAXCYC= 25 DELTAC=.0000
0DISTANCE MATRIX
1
2
3

1
2
3
.0000 .9900 .9900
.9900 .0000 1.5656
.9900 1.5656 .0000

1
2
3
1 .0000 1.0063 1.0063
2 1.0063 .0000 .0345
3 1.0063 .0345 .0000
2.04704595
0SPIN= 0

ENERGY LEVELS (EV)
E( 1) = 14.73419 0
E( 2) =

3.38040 0

E( 3) = -13.62000 2
E( 4) = -14.43845 2
E( 5) = -16.97488 2
E( 6) = -31.03678 2
ENERGY= -150.09316161 EV.

0

4 ORBITALS FILLED
0 HALF FILLED
0WAVE FUNCTIONS
1
2
3
4
5
6

1
-.9712
.0000
.0000
.5517
.8029
.8029

2
.0000
.9247
.0000
.0000
-.8588
.8588

3
4
5
.0000 .2319 .0000
.0000 .0000 -.6834
1.0000 .0000 .0000
.0000 .9213 .0000
.0000 -.1291 -.3815
.0000 -.1291 .3815

6
.7793
.0000
.0000
.0224
.2029
.2029

REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM
1
2
3
1 5.9549 .6598 .6598
2 .6598 .4068 -.0882
3 .6598 -.0882 .4068
1
2
3

1
2
3
4
5
6
.6344 .7508 2.0000 1.8608 1.2492 1.5048
.6828 .6246 .0000 .0696 .3754 .2476
.6828 .6246 .0000 .0696 .3754 .2476

0ATOM
O 1
H 2
H 3

ROWS

NET CHG.
-.61478
.30739
.30739

S
X
Y
Z
X2-Y2 Z2
XY
XZ
YZ
1.56003 1.24916 2.00000 1.80558
.69261
.69261

4 0 0 0
NHHH
2 1.6237-17.830
2 1.6170-12.040
0 0.0000 0.000
0.00000.00000.0000
7 30 2 3 0
1 1.2000-13.600
0 0.0000 0.000
0 0.0000 0.000
0.00000.00000.0000
1 10 1 0 0
1 1.2000-13.600
0 0.0000 0.000
0 0.0000 0.000
0.00000.00000.0000
1 10 1 0 0
1 1.2000-13.600
0 0.0000 0.000
0 0.0000 0.000
0.00000.00000.0000
1 10 1 0 0
0.00000000 0.00000000
1 2 1.22000
60.00000
1 3 1.22000
180.00000
1 4 1.22000
60.00000
0 1 2 1 0.010 1.5000

0.00000000
0.00000
90.00000
180.00000

ATOM
D
N
H
H
H

1
2
3
4

X

.00000
1.10000
.00000
-1.10000

Y
.00000
.00000
.00000
.00000

Z
.00000
-.60000
1.20000
-.60000

N EXP

S

COUL

N EXP

P

COUL

D
CONTRACTED D
N EXPD1 COUL C1 C2 EXPD2 AT EN S P

2 1.6237 -17.8300 2 1.6170 -12.0000 0
1 1.2000 -13.6000 0 .0000 .0000 0
1 1.2000 -13.6000 0 .0000 .0000 0
1 1.2000 -13.6000 0 .0000 .0000 0

0DISTANCE MATRIX
1
2
3
4
1 .0000 1.2200 1.2200 1.2200
2 1.2200 .0000 2.1131 2.1131
3 1.2200 2.1131 .0000 2.1131
4 1.2200 2.1131 2.1131 .0000

.0000
.0000
.0000
.0000

.0000 .00000 .00000 .00007 30 2 3 0
.0000 .00000 .00000 .0000 1 10 1 0 0
.0000 .00000 .00000 .0000 1 10 1 0 0
.0000 .00000 .00000 .0000 1 10 1 0 0

ENERGY LEVELS (EV)

E( 1) = 16.00000 0
E( 2) = 5.00000 0
E( 3) = 5.00000 0
E( 4) = -12.00000 2
E( 5) = -17.00000 2
E( 6) = -17.00000 2
E( 7) = -23.00000 2
ENERGY= -131.90074095 EV.
0WAVE FUNCTIONS
1
2
3
4
5
6
7
1 -1.2702 .0000 .0000 .0000 .0000 .0000 .6009
2 .0000 .0000 1.0340 .0000 -.5499 .0000 .0000
3 .0000 .0000 .0000 -1.0000 .0000 .0000 .0000
4 .0000 1.0340 .0000 .0000 .0000 .5499 .0000
5 .6891 .1677 -.9271 .0000 -.4096 -.2804 .2515
6 .6891 -.8867 .0000 .0000 .0000 .4949 .2515
7 .6891 .7190 .6088 .0000 .4476 -.2145 .2515
1
2
3
4
1 3.9387 .7613 .7613 .7612
2 .7613 .6193 -.0267 -.0267
3 .7613 -.0267 .6192 -.0267
4 .7612 -.0267 -.0267 .6192
0ATOM NET CHG.
S
X
Y
Z
X2-Y2 Z2
XY
XZ
YZ
N 1 -.08064
1.12890 .97589 2.00000 .97586
H 2
.02683
.97317
H 3
.02689
.97311
H 4
.02693
.97307

ATOM

X

Y

Z

S

D
CONTRACTED D
N EXP COUL
N EXP COUL
N EXPD1 COUL C1 C2 EXPD2 AT EN S P D
F 1 .00000 .00000 .00000 2 2.5630 -37.5800 2 2.5500 -17.4200 0 .0000 .0000 .00000 .00000 .0000 9 40 2 5 0
H 2 .00000 .00000 -1.20000 1 1.2000 -13.6000 0 .0000 .0000
0 .0000 .0000 .00000 .00000 .0000 1 10 1 0 0
0CHARGE= 0 IEXIT= 0 ITYPE= 0 IPUNCH= 0 CON= 2.250 AU= .52920 SENSE= 2.000 MAXCYC= 25 DELTAC= .00000100
0DISTANCE MATRIX
1
2
1 .0000 1.2000
2 1.2000 .0000
1
2
1 .0000 .0846
2 .0846 .0000
.08463978
0SPIN= 0
0
ENERGY LEVELS (EV)
E( 1) = -5.49648 0
E( 2) = -17.42000 2
E( 3) = -17.42000 2
E( 4) = -18.31403 2
E( 5) = -38.02579 2
ENERGY= -182.27500191 EV.
4 ORBITALS FILLED
0 HALF FILLED
0WAVE FUNCTIONS
0

1
2
3
4
5
1 -.4135 .0000 .0000 -.1327 .9547
2 .0000 .0000 -.7071 .0000 .0000
3 .0000 -.7071 .0000 .0000 .0000
4 .5194 .0000 .0000 -.8947 .0126
5 1.0364 .0000 .0000 .2825 .1295

P

1
2
1 7.4596 .3472
2 .3472 .1931
1
2
3
4
5
1 .3667 2.0000 2.0000 1.7383 1.8949
2 1.6333 .0000 .0000 .2617 .1051
0ATOM NET CHG.
OCCUPATION
F 1
H 2

-.63325
.63325

ATOMIC ORBITAL OCCUPATION

ROWS

FOR GIVEN MO

S
X
Y
Z
X2-Y2
1.90871 2.00000 2.00000 1.72454
.36675

Z2

XY

XZ

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Quantum course