Quantum course

By: Prof .Dr \ Mohamed KhaledBy: Prof .Dr \ Mohamed Khaled

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.

Quantum Chemistry

Physical Chemistry Inorganic Chemistry Organic Chemistry Photochemistry Polymer Surface and Catalysis Drug DesignToxicity

Historical background of quantum mechanics:Nature of light:

Hertz, 1888, has showed that light is electromagnetic waves. = c /

where, is the wavelength, c is the speed of light =2.998 x 1010 cm/sec, is the frequency cm/sec.Max Plank has assumed only certain quantities of light energy (E) could be emitted. E = hWhere, 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 = hh = 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 -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, = E / hwhere, 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. = h / mv = h / pwhere, 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 / 2

x . px h / 4

Werner HeisenbergNobel prize 1932

To describe the state in quantum mechanics, we postulate the existence of a function of the

coordinates called the wave function (State function), .

= (x, t)

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

Wave Function:

What (x) means?

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

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

- has no physical meaning but * is the probability of locating the electron at a

given position.

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

1)()( * zyxxx

If we have two different wave functions, 1 and 2 will be Normalized function when:

1)()( 1*

1 zyxxx and

1)()( 2*

2 xx

But if

0)()( 21 xx or

0)()( 2*

1* xx

The function is called orthogonal function.

But if

ijji xx )()(

Where, ij (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), . For one particle, one-dimensional system:

= (x, t)

It contains all information about a system. The probability of finding a particle in a

given place can be given by (Probability description).

Born postulates | (x,t) |2 dx is the probability of finding a particle at position x and

at time t ( Probability density ). must satisfy Schrödinger equation. As t passes, changes to differential equation:

where, i= , m = particle mass, V(x,t) = potential energy. This is called Time-dependent Schrödinger equation.

),(),(),(

2

),(2

22

txtxVx

tx

mt

tx

i

1

Erwin SchrödingerNobel prize 1933

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

the partial derivatives of this equation:

)()(),( txtx

)()(),(

)()(),(

2

2

2

2

tx

x

x

tx

xt

t

t

tx

Making the substitution in equation 2:

)()()()(

)(

2)(

)(2

22

txxVtx

x

mx

t

t

i

Dividing by )()( tx

)()(

)(

1

2

)(

)(

12

22

xVx

x

xmt

t

ti

Taking the left side of equation (3): t

iE

t

t

)(

)(

On integration: ln CiEt

t

)( C is a constant of integration

iEt

iEtc

At

t

)(

)(

iEt

t

)(

iEtxtx

)(),(

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

*2 where, * is a complex conjugate of

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

imaginary value.

2**0

*2

)()()()()(

)()(),(

xxxxx

xxtxiEtiEt

for stationary state

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

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

)()()()(

2 2

22

xExxVx

x

m

or 0)()(8)(

2

2

2

2

xxVE

h

m

x

x

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.

)()()(

2 2

22

xExxVxm

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.

)()()(

2 2

22

xExxVxm

This operator is called the Hamiltonian operator for the system.

Kinetic energy = 2

22 ˆ

2 xm

Ĥ = Kinetic energy + Potential energy

= )(ˆ

2 2

22

xVxm

So, the Eigen value equation:

Ĥ (x) = E (x)

-A particle in one-dimensional box:

V

I V = 0 III

II Ψ = 0

X = 0 X = 1 X

)()()(

2 2

22

xExx

x

m

0)()(2)(

22

2

xE

m

x

x

)(

)(2

2

xx

x

)(

1)(2

2

xx

x

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

I(x) = zero 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

0)(

2)(22

2

xE

m

x

x

l

xn

lx

sin2

)(

2

22

8ml

hnE

n= 1,2,3,………

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

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

x 0 a -a

E Velocity=0

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= -kxWhere, F is a restoring force, k is a force constant, and x is a displacement on x-axis.

F = x

V

= - kx

By integration: V(x) = ½ kx2

m

k

2

1

2222 22/1)( mxkxxV

222

22

2222

22

22

22

2

22

2/12

ˆ

xxm

mxxm

kxxm

H

Where, /2 m

The Schrödinger equation H (x) = E (x), after multiplication on by 22

m

0)(2)( 222

2

2

xxmE

x

x

2

2x

A

24

1

0

2x

224

1

2

24

13

1

2

2

124

4

x

x

x

x

The energy of a harmonic oscillator is quantized.

En = (n + ½ ) h 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:

21

04

eé

where, e is the proton charge in coulomb. To deal with the internal motion of the system, we introduce as the mass of the particle. = me mn / me + mn

where, me and mn are the electronic and nuclear masses.

r

eZH

22

2

ˆ2

ˆ

where, 2 is Laplacian operator: 2

2

2

2

2

22ˆ

zyx

So, the time-independent Schrödinger equation is:

E

r

eZ

22

2

ˆ2

x

y

z

r

me

mn

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

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.

),()()( YrRx

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

)

2()

2(

1)(2

1)1()

2()(

0

12)

2(

0

21

30

0

na

ZrL

na

Zr

lnn

ln

na

ZrR l

lnna

Zr

lln

imm

lm

l Pml

mllY

)(cos

1(

)1(

4

12),(

21

.

.

5292.0

2

2

0 e

a

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

a

Zr

a

Zr

a

Z

a

ZrR

23

100

23

10

1

2)(

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

22

42

2 hn

eZE

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)

Probability densities for some hydrogen-atom states

Shapes of electron cloud:

Positive (Bonding)

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

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

Sij = ∫i j dτ Three different kinds of overlap are shown in Fig. ( 9).

σ and π bonds

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 H X

Y

Table 5

1 2 3 4 5

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

2 0 0 0 N N 0.000000 .000000 .0000002 1.9237-20.3302 1.9170-14.5400 0.0000 00.000 .0000 .00000.0000 7 30 2 0 3 1.020000 0.000000 0.0000002 1.9237-20.3302 1.9170-14.5400 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 30CHARGE= 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 20 ENERGY= -187.27312236 EV. 5 ORBITALS FILLED 0 HALF FILLED0WAVE FUNCTIONS0MO'S IN COLUMNS, AO'S IN ROWS

1 2 3 4 5 6 7 8 1 .0000 .0000 .0000 .0000 .0000 .0000 .5068 .5223 2 -1.2989 .0000 .0000 .0000 .0000 .5144 .0000 .0000 3 .0000 -.7443 .0000 .0000 .5234 .0000 .0000 .0000 4 .0000 .0000 .7443 .5234 .0000 .0000 .0000 .0000 5 .0000 .0000 .0000 .0000 .0000 .0000 -.5068 .5223 6 -1.2989 .0000 .0000 .0000 .0000 -.5144 .0000 .0000 7 .0000 .7443 .0000 .0000 .5234 .0000 .0000 .0000 8 .0000 .0000 -.7443 .5234 .0000 .0000 .0000 .0000

0REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM 1 2 1 4.0437 2.7000 2 2.7000 4.04370REDUCED 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 NET CHG. ATOMIC ORBITAL OCCUPATION FOR GIVEN MO OCCUPATION S X Y Z X2-Y2 Z2 XY XZ YZ

N 1 .00000 1.40947 1.59053 1.00000 1.00000 N 2 .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 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 1.02000 .00000 .00000 2 1.9237 -20.3300 2 1.9170 -14.5400 0 .0000 .0000 .00000 .00000 .0000 7 30 2 0 30CHARGE= 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 1.0200 2 1.0200 .0000


1 2 1 .0000 4.4351 2 4.4351 .0000 4.43514361

0SPIN= 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


1 2 3 4 5 6 7 8 1 -1.6157 .0000 .0000 .3190 -.4437 .0000 .0000 .5223 2 -1.2989 .0000 .0000 -.6477 .5144 .0000 .0000 .1333 3 .0000 -.7443 -.4460 .0000 .0000 .3426 .5068 .0000 4 .0000 -.4460 .7443 .0000 .0000 .5068 -.3426 .0000 5 1.6157 .0000 .0000 .3190 .4437 .0000 .0000 .5223 6 -1.2989 .0000 .0000 .6477 -.5144 .0000 .0000 -.1333 7 .0000 .7443 -.4460 .0000 .0000 .3426 .5068 .0000 8 .0000 .4460 -.7443 .0000 .0000 .5068 -.3426 .0000

REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM

1 2 1 4.0437 1.9126 2 1.9126 4.04370REDUCED 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 NET CHG. ATOMIC ORBITAL OCCUPATION FOR GIVEN MO OCCUPATION S X Y Z X2-Y2 Z2 XY XZ YZ

N 1 .00000 1.40947 1.59053 1.00000 1.00000 N 2 .00000 1.40947 1.59053 1.00000 1.00000

3 0 0 0 O H H 2 2.2459-28.4802 2.2266-13.6200 0.0000 0.0000.00000.00000.0000 8 35 2 4 01 1.2000-13.6000 0.0000 0.0000 0.0000 0.0000.00000.00000.0000 1 10 1 0 01 1.2000-13.6000 0.0000 0.0000 0.0000 0.0000.00000.00000.0000 1 10 1 0 0 0.00000000 0.00000000 0.00000000 1 2 0.99000 52.25000 0.00000 1 3 0.99000 52.25000 180.00000 0 1 2 2 1.000 132.0000

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 00CHARGE= 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 .0000 .9900 .9900 2 .9900 .0000 1.5656 3 .9900 1.5656 .0000


1 2 3 1 .0000 1.0063 1.0063 2 1.0063 .0000 .0345 3 1.0063 .0345 .0000 2.047045950SPIN= 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 20 ENERGY= -150.09316161 EV.

4 ORBITALS FILLED 0 HALF FILLED0WAVE FUNCTIONS0MO'S IN COLUMNS, AO'S IN ROWS

1 2 3 4 5 6 1 -.9712 .0000 .0000 .2319 .0000 .7793 2 .0000 .9247 .0000 .0000 -.6834 .0000 3 .0000 .0000 1.0000 .0000 .0000 .0000 4 .5517 .0000 .0000 .9213 .0000 .0224 5 .8029 -.8588 .0000 -.1291 -.3815 .2029 6 .8029 .8588 .0000 -.1291 .3815 .2029

REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM

1 2 3 1 5.9549 .6598 .6598 2 .6598 .4068 -.0882 3 .6598 -.0882 .40680REDUCED CHARGE MATRIX, MO'S IN COLUMNS, ATOMS IN ROWS

1 2 3 4 5 6 1 .6344 .7508 2.0000 1.8608 1.2492 1.5048 2 .6828 .6246 .0000 .0696 .3754 .2476 3 .6828 .6246 .0000 .0696 .3754 .2476 0ATOM NET CHG. ATOMIC ORBITAL OCCUPATION FOR GIVEN MO OCCUPATION S X Y Z X2-Y2 Z2 XY XZ YZ

O 1 -.61478 1.56003 1.24916 2.00000 1.80558 H 2 .30739 .69261 H 3 .30739 .69261

4 0 0 0 N H H H 2 1.6237-17.8302 1.6170-12.0400 0.0000 0.0000.00000.00000.0000 7 30 2 3 01 1.2000-13.6000 0.0000 0.0000 0.0000 0.0000.00000.00000.0000 1 10 1 0 01 1.2000-13.6000 0.0000 0.0000 0.0000 0.0000.00000.00000.0000 1 10 1 0 01 1.2000-13.6000 0.0000 0.0000 0.0000 0.0000.00000.00000.0000 1 10 1 0 0 0.00000000 0.00000000 0.00000000 1 2 1.22000 60.00000 0.00000 1 3 1.22000 180.00000 90.00000 1 4 1.22000 60.00000 180.00000 0 1 2 1 0.010 1.5000

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 1.6237 -17.8300 2 1.6170 -12.0000 0 .0000 .0000 .00000 .00000 .00007 30 2 3 0 H 2 1.10000 .00000 -.60000 1 1.2000 -13.6000 0 .0000 .0000 0 .0000 .0000 .00000 .00000 .0000 1 10 1 0 0 H 3 .00000 .00000 1.20000 1 1.2000 -13.6000 0 .0000 .0000 0 .0000 .0000 .00000 .00000 .0000 1 10 1 0 0 H 4 -1.10000 .00000 -.60000 1 1.2000 -13.6000 0 .0000 .0000 0 .0000 .0000 .00000 .00000 .0000 1 10 1 0 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

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 FUNCTIONS0MO'S IN COLUMNS, AO'S IN ROWS 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 .25150REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM 1 2 3 4 1 3.9387 .7613 .7613 .7612 2 .7613 .6193 -.0267 -.0267 3 .7613 -.0267 .6192 -.0267 4 .7612 -.0267 -.0267 .61920ATOM NET CHG. ATOMIC ORBITAL OCCUPATION FOR GIVEN MO OCCUPATION 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 P 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 00CHARGE= 0 IEXIT= 0 ITYPE= 0 IPUNCH= 0 CON= 2.250 AU= .52920 SENSE= 2.000 MAXCYC= 25 DELTAC= .000001000DISTANCE MATRIX 1 2 1 .0000 1.2000 2 1.2000 .00000TWO BODY REPULSION ENERGY MATRIX 1 2 1 .0000 .0846 2 .0846 .0000 .084639780SPIN= 00 ENERGY LEVELS (EV) E( 1) = -5.49648 0

E( 2) = -17.42000 2

E( 3) = -17.42000 2

E( 4) = -18.31403 2


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

0REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM 1 2 1 7.4596 .3472 2 .3472 .1931

0REDUCED CHARGE MATRIX, MO'S IN COLUMNS, ATOMS IN ROWS 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. ATOMIC ORBITAL OCCUPATION FOR GIVEN MO OCCUPATION S X Y Z X2-Y2 Z2 XY XZ YZ F 1 -.63325 1.90871 2.00000 2.00000 1.72454 H 2 .63325 .36675

Quantum course

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