1

ab-initio calculation of structural properties

of cubic ice

Hadi Arabi

Maryam VahediPhysics Department

Birjand University, Birjand, Iran

Ali mokhtariDepartment of physics Isfahan University of Technology

Isfahan Iran

1

1

Outline

1-Different phases of ice

2-Theoretical aspect of the work

3- Data analysis

4- Conclusions

1

The most important matter in the world is water. 15% of the weight of Earth is water , and in the North and South ocean most of them is in the form of ice. Therefore physical and chemical properties, of different phases of ice are great important.

1

Diagram of various phases of ice

1

Table of different phases of ice with their relative parameters

1

General ice with hexagonal structure (IceIh) between -80 up to 0 centigrade and general pressure

1

Structure of cubic ice between -130 up to -80 centigrade which is unstable . This phasecan be produced from phases II ,III ,VI,VII in liquid nitrogen temperature. In -100the cubic ice changes to Ih phase of ice.

1

Unit cell of cubic ice which shows only oxygen atoms . O-H-O bands are represented with orange lines

1

Relation between unit cell and layer structure in closed-packed of cubic ice

1Ice- II phase with rhombohedral structure

1

Cubic III-phase structure, oxygen atoms shows with blue circle which closer oxygen with bigger circle and O-H-O bands represented as red lines

1

Cubic V-phase with monoclinic structure. The unit cells are represented as cubic

1

Cubic VII phase composes of two structures of cubic Ice which represented one with red lines and the other with orange lines.

The conventional cell has 8 bcc cubic.

1

Cubic Ice

In this structure eight molecules of water are in the conventional cell.Number of configurations are 2**16.Bernell-Fowler rules are restricted the number of states.1) The distance between each oxygen and hydrogen atoms in a molecule is about 0.95 angstrom.2) Water molecule in unite cell stay in a position where only one hydrogen of each molecule is located between two oxygen atoms.3) Each molecule of water has surrounded by for other oxygen atoms. It is located somehow that always two hydrogen atoms pointed to two oxygen atoms.

1

1

Total dipole moments of each configurations of cubic ice structures which are 8,4√√√√2,4 and 0 respectively.

1

The important of computational physics in solid state physics

1- Analysis of physical phenomena which are not possible to happen or if they happen, it is not possible to repeat them.2-Variation of parameters in a physical process in the direction and size of variation which are not normal.

Classic3-Many body system

Quantum

1

Born-Oppenhimer approximation

Nucleus in atoms are stable regarding to the motion of electrons

),...,,,,...,,(),...,,,,...,,( 21212121 αα ψψ RRRrrrERRRrrrH NN

������ =

nnetot EEE +=

NNeeNeNk

ek HHHHHH −−− ++++=

)()()( ieiee

eek rErHHH ψψ =++ −

)()()( αα ψψ RERHHH totalNN

eNk =++ −

1

Solution of Schrodinger equation for system of electrons

First we have to change equation of system of electrons to a sum of separate equations for each electron. There are two approaches for it:

1- Wave function is accepted as a variable parameter: Hartree and Hartree-Fock methods.

2- Density is considered as a variable parameter : Tomas Fermi and density functional theory

1

Hartree and Hartlee-Fock method:In this method the wave function of the system is considered as a multiplication of wave functions of single particles. Also we consider that each electron see an effective potential from nucleus and other electrons

)()...()(),...,,( 21321 Nrrrrrr������ ψψψψ =

)()()2

( 22

iiiiiext rrVm

��� ψεψ =+∇−

� � ′−′′

+−

−=α α

α

rr

rnrde

rr

ZerV

iext ��

�

��)(

)( 22

)()( �=i

ii rrn�� ψ

1

Density functional theory

This theory in 1964 presented by Kohn and Hohonberg.We can deduced all properties of a system in ground state using density of ground state.

Kohn and Sham in 1965 replace the equation of the system by groups of equations of single particles in the system.

� �� +−

′++= )]([)()(

2)()()]([)]([

2

rnErr

rnrnrdrd

edrrVrnrnTrnE XC

���

�������

1

Problems with single particle equationsVery large number of electrons in system:

Using Bloch theorem and choosing appropriate K in reciprocal lattice.

Wave function of a particle in level n with wave vector of K

)().exp()(,,

rurKirKnKn

������ =ψ )()(

,,Rruru

KnKn

����� +=

).exp()()(,

rGiGKcru

G

nKn

�����

�

� �′

+=

)]).(exp[)()(, rGKiGKcr

G

nkn

������

�

++=�′

ψ

1

Pseudo potential approximation

Area around the nucleus is divided to core and valance regions

Wave function of core electrons are localized in the region and play no role in ionic bonds.

Valance electrons in core region have high kinetic energy and their wave function has a lot of vibrations in the core which need a huge number of plane wave for their construction.

In summation plane wave those with low kinetic energy has more important. Therefore we can continue the extension of plane waves up to a certain value called cut off energy.

1

Wave function,pseudo wave function, real potential and pseudo potential are illustrated qualitatively

1

Properties of ad-initio pseudo potentials

Wave function and pseudo wave function of conduction valance band must coincide to each other out of core. Also potential andpseudo potential functions have the same properties.

Pseudo potential function is not localized outside core, which means it relates to L quantum number. We have to consider the charge conservation for both potential

and pseudo potentials inside the core.

�� =cc r

psl

r

l drrrdrrr

0

22

0

22)(4)(4�� ψπψπ

1

Wave scattering form core for potential and pseudo potential should be the same. This condition leads to continues of logarithmic derivative of wave function and pseudo wave function at r = rc

1

Kerker method

Pseudo wave function in the core region explained as:

)()()( 1 rfrrRrrF ll

l

���� ===

)()( rPrf ll

�� =)()( rP

llerf�� =

lllll rrrrP δγβα +++= 234)(�

1

0))r(Pexp(r]E)r(Vr

)1l(l

dr

d[ l

1ll

psl22

2

=−+++− + ��

ccrrlrrl

1l )r())r(Pexp(r ==+ ψ=

��

])(

[)( 1+= lc

clcl

r

rLnrP

�� ψ

ccrrlrrl

ll

l rrPrrPrl==

+ ′=′++ )()(exp(])()1[( 1 ��� ψ

ccrrlrrl

lll

ll

l rrPrPrprrPlrl==

++− ′′=′+′′+′+++ )())(exp(])()1(2)1([ 1211 ��� ψ

1

Qc-tuning method This method is an extended of optimum procedure for producing pseudo potential.

Rap and his co-worker,1990,presented a method called RRKJ. They choose the pseudo wave function of valence electrons as a linear combination of Bessel equation. First we produce the pseudo wave function and then by using reciprocal schrodinger equation, we calculate the pseudo potential.

�=

=n

i

ilipsl rqjr

1

)()(�� αψ

crr <

)()( rr lpsl

�� ψψ = crr ≥

1

Wave function and pseudo wave function of S,P and D orbital of oxygen atom produced by Kerker method.

0 1 2 3 4 5 6 7 8 9 10

rΨΨ ΨΨ

(r)

-1.0-0.8-0.6-0.4

-0.20.00.20.40.60.81.0

Oxygen Atom

0 1 2 3 4 5 6 7 8 9 10

r ΨΨ ΨΨ(r

)

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

R(Ao)

0 1 2 3 4 5 6 7 8 9 10

r ΨΨ ΨΨ(r

)

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

Kerker Pseudopotential

s

p

d

1Potential and pseudo potential of S,P and D orbitals of oxygen atom produced by Kerker method.

Kerker pseudopotential

0 1 2 3 4 5 6 7 8 9 10

Pse

udop

oten

tial

-30

-25

-20

-15

-10

-5

0

R(Ao)

Oxygen Atom

d

p

s

1

Logarithmic derivative of real potential and pseudo potential ofS,P and D orbitals of oxygen perused

..........PS------- ACTUAL

Kerker Pseudopotential

-4 -3 -2 -1 0 1 2 3 4

Log

arit

hmic

Der

ivat

ive

-5-4-3-2-1012345

-4 -3 -2 -1 0 1 2 3 4

Log

arit

hmic

Der

ivat

ive

-5

-4

-3-2

-101

23

45

-4 -3 -2 -1 0 1 2 3 4

Log

arit

hmic

Der

ivat

ive

-5

-4

-3

-2

-1

0

1

2

3

4

5

Energy (ev)

Oxygen Atom

s

p

d

1

(Left)-Wave function and pseudo wave function of S,P orbitals of hydrogen atom.

(right)- Potential and pseudo potential of S and P orbitals of hydrogen atom.

0 1 2 3 4 5 6 7 8 9 10

rΨΨ ΨΨ

(r)

-1.0

-0.8-0.6-0.4-0.20.00.20.40.60.81.0

Hydrogen Atom

0 1 2 3 4 5 6 7 8 9 10

r ΨΨ ΨΨ(r

)

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

Kerker pseudopotential

s

p

R(Ao)

1

Logarithmic derivative of real potential and pseudo potential of S and P orbitals of hydrogen

..........PS------- ACTUAL

Kerker Pseudopotential

-4 -3 -2 -1 0 1 2 3 4

Log

arit

hmic

Der

ivat

ive

-10

-5

0

5

10

-4 -3 -2 -1 0 1 2 3 4

Log

arit

hmic

Der

ivat

ive

-10

-5

0

5

10

Energy (ev)

Hydrogen Atom

s

p

1

(Left)-Wave function and pseudo wave function of S,D orbitals of hydrogen atom. (right)- Potential and pseudo potential of S and D orbitals of hydrogen atom.

Kerker Pseudopotential

0 1 2 3 4 5

Pse

udop

oten

tial

-12

-10

-8

-6

-4

-2

0

_____s

........... d

R(Ao)

Hydrogen Atom

s

d

0 1 2 3 4 5 6 7 8 9 10

rΨΨ ΨΨ

(r)

-1.0-0.8

-0.6-0.4-0.20.00.20.40.6

0.81.0

Hydrogen Atom

R(Ao)

0 1 2 3 4 5 6 7 8 9 10

r ΨΨ ΨΨ(r

)

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

Kerker Pseudopotential

s

d

1

Logarithmic derivative of real potential and pseudo potential of S and P orbitals of hydrogen.

.� .� .� .� .� .� .� .� .� .�P� S�-� -� -� -� -� -� -� �A� C� T� U� A� L� K� e� r� k� e� r� �P� s� e� u� d� o� p� o� t� e� n� t� i� a� l�

-�4� -�3� -�2� -� 1� 0� 1� 2� 3� 4�

L�o�g�a�r�i�t�h�m�i�c� � �D�e�r�i�v�a�t�i�v�e�

-�1� 0�

-� 5�

0�

5�

1� 0�

-� 4� -�3� -�2� -�1� 0� 1� 2� 3� 4�

L�o�g�a�r�i�t�h�m�i�c� � �D�e�r�i�v�a�t�i�v�e�

-�1� 0�

-�5�

0�

5�

1� 0�

E� n� e� r� g� y� �(� e� v� )�

H� y� d� r� o� g� e� n� �A� t� o� m�

s�

d�

1

(Left)-Wave function and pseudo wave function of S,P orbitals of oxygen atom.

(right)- Potential and pseudo potential of S and d orbitals of hydrogen atom.

Qc-Tuning Pseudopotential

0 1 2 3 4 5

PSE

UD

OP

OT

EN

TIA

L

-7

-6

-5

-4

-3

-2

-1

0

_____s

---------p

.........d

R(A0)

Hydrogen Atom

s

d

0 1 2 3 4 5 6 7 8 9 10

-1.0

-0.8-0.6

-0.4-0.20.0

0.20.4

0.60.81.0

Oxygen Atom

0 1 2 3 4 5 6 7 8 9 10

r ΨΨ ΨΨ (r

)

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

R (Ao)

0 1 2 3 4 5 6 7 8 9 10

r ΨΨ ΨΨ (r

)

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

Qc-Tuning Pseudopotential

S

p

d

r ψψ ψψ(( ((r

)

1

Logarithmic derivative of real potential and pseudo potential of S and P orbitals of oxygen perused by Qc-tuning method

..........PS

------- ACTUALQc-Tuning Pseudopotential

-4 -3 -2 -1 0 1 2 3 4

Log

arit

hmic

Der

ivat

ive

-5-4-3-2-1012345

-4 -3 -2 -1 0 1 2 3 4

Log

arit

hmic

Der

ivat

ive

-5

-4

-3

-2-1

0

1

2

3

4

5

-4 -3 -2 -1 0 1 2 3 4L

ogar

ithm

ic D

eriv

ativ

e

-5

-4

-3

-2

-1

0

1

2

3

4

5

Energy (ev)

Oxygen Atom

1

(Left)-Wave function and pseudo wave function of S,P orbitals of hydrogen atom.(right)- Potential and pseudo potential of S and d orbitals of hydrogen atom.

Qc-Tuning Pseudopotential

0 1 2 3 4 5

Pse

udop

oten

tial

-6

-5

-4

-3

-2

-1

0

_____s

---------p

R (Ao)

Hydrogen Atom

s

p0 1 2 3 4 5 6 7 8 9 10

rΨΨ ΨΨ

(r)

-1.0-0.8-0.6-0.4-0.20.00.20.40.6

0.81.0

Hydrogen Atom

0 1 2 3 4 5 6 7 8 9 10

r ΨΨ ΨΨ(r

)

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

Qc-Tuning Pseudopotential

s

p

R(Ao)

1

Logarithmic derivative of real potential and pseudo potential of S and P orbitals of hydrogen.

..........PS------- ACTUAL

Qc-Tuning Pseudopotential

-4 -3 -2 -1 0 1 2 3 4

Log

arith

mic

Der

ivat

ive

-5-4-3-2-1012345

-4 -3 -2 -1 0 1 2 3 4

Log

arit

hmic

Der

ivat

ive

-5

-4

-3

-2

-1

0

1

2

3

4

5

Energy (ev)

Hydrogen Atom

s

p

1

(Left)-Wave function and pseudo wave function of S,d orbitals of hydrogen atom.(right)- Potential and pseudo potential of S and d orbitals of hydrogen atom.

Qc-Tuning Pseudopotential

0 1 2 3 4 5

PS

EU

DO

PO

TE

NT

IAL

-7

-6

-5

-4

-3

-2

-1

0

_____s

---------p

.........d

R(A0)

Hydrogen Atom

s

d

0� 1� 2� 3� 4� 5� 6� 7� 8� 9� 1� 0�

�r�ΨΨΨΨ�(�r�)�

-� 1� .� 0�-� 0� .� 8�-� 0� .� 6�-� 0� .� 4�-� 0� .� 2�0� .� 0�0� .� 2�0� .� 4�0� .� 6�0� .� 8�1� .� 0�

H� y� d� r� o� g� e� n� �A� t� o� m�

R� (� A� 0� )�

0� 1� 2� 3� 4� 5� 6� 7� 8� 9� 1� 0�

r�ΨΨΨΨ�(�r�)�

0� .� 0�

0� .� 2�

0� .� 4�

0� .� 6�

Q� c� -� T� u� n� i� n� g� �P� s� e� u� d� o� p� o� t� e� n� t� i� a� l�

s�

d�

1

Logarithmic derivative of real potential and pseudo potential of S and D orbitals of hydrogen.

..........PS------- ACTUAL

Qc-Tuning Pseudopotential

-4 -3 -2 -1 0 1 2 3 4

Log

arit

hmic

Der

ivat

ive

-10

-5

0

5

10

-4 -3 -2 -1 0 1 2 3 4

Log

arit

hmic

Der

ivat

ive

-10

-5

0

5

10

Energy (ev)

Hydrogen Atom

s

d

1

For minimizing the energy of each configuration and structural parameter identification, we do need to determine the positions of atoms and use them as an input for calculation

1

0.29160 0.29160 0.308270.95832 0.95832 0.308270.04168 0.20840 0.058350.20840 0.04168 0.058350.45832 0.45832 0.308270.79160 0.79160 0.308270.70840 0.54168 0.058350.54168 0.70840 0.058350.54168 0.20840 0.558350.70840 0.04168 0.558350.79160 0.29160 0.808270.45832 0.95832 0.808270.20840 0.54168 0.558350.04168 0.70840 0.558350.95832 0.45832 0.808270.29160 0.79160 0.808270.12504 0.12504 0.141710.62504 0.62504 0.141710.62504 0.12504 0.641710.12504 0.62504 0.641710.37496 0.37496 0.391630.87496 0.87496 0.391630.87496 0.37496 0.891630.37496 0.87496 0.89163

0.29160 0.29160 0.308270.95832 0.95832 0.308270.04168 0.20840 0.058350.20840 0.04168 0.058350.45832 0.45832 0.308270.70839 0.70839 0.225060.79160 0.45831 0.974990.54168 0.70840 0.058350.54168 0.20840 0.558350.79160 0.95831 0.474980.70839 0.20839 0.725060.45832 0.95832 0.808270.20840 0.54168 0.558350.04168 0.70840 0.558350.95832 0.45832 0.808270.29160 0.79160 0.808270.12504 0.12504 0.141710.62504 0.62504 0.141710.62504 0.12504 0.641710.12504 0.62504 0.641710.37496 0.37496 0.391630.87496 0.87496 0.391630.87496 0.37496 0.891630.37496 0.87496 0.89163

0.29160 0.29160 0.308270.04168 0.04168 0.225060.95831 0.29160 0.974980.20840 0.04168 0.058350.45831 0.45832 0.308270.79160 0.79160 0.308270.7084 0.5417 0.058350.5417 0.7084 0.058350.5417 0.2084 0.558350.79160 0.95831 0.474980.70839 0.20839 0.725060.45832 0.9583 0.80830.20840 0.54168 0.558350.04168 0.70840 0.558350.95832 0.45832 0.808270.29160 0.79160 0.808270.12504 0.12504 0.141710.62504 0.62504 0.141710.62504 0.12504 0.641710.12504 0.62504 0.641710.37496 0.37496 0.391630.87496 0.87496 0.391630.87496 0.37496 0.891630.37496 0.87496 0.89163

0.29160 0.29160 0.308270.95832 0.95832 0.308270.04168 0.20840 0.058350.20840 0.04168 0.058350.54168 0.54168 0.225060.70839 0.70839 0.225060.79160 0.45831 0.974980.4583 0.79160 0.974980.4583 0.29160 0.474980.79160 0.95831 0.474980.70839 0.20839 0.725060.5417 0.04168 0.725060.20840 0.54168 0.558350.04168 0.70840 0.558350.95832 0.45832 0.808270.29160 0.79160 0.808270.12504 0.12504 0.141710.62504 0.62504 0.141710.62504 0.12504 0.641710.12504 0.62504 0.641710.37496 0.37496 0.391630.87496 0.87496 0.391630.87496 0.37496 0.891630.37496 0.87496 0.89163

1 2 3 4

Positions of atoms in four different configurations of cubic ice

1

Total energy versus cut off energy with Qc-tuning method and GGA approximation

Total energy versus cut of energy obtained by Qc-tuning and GGA approximation

(unit cell) (conventional cell)

Ecut(ev)200 400 600 800 1000 1200 1400

Tota

l Ener

gy(

ev)

-940

-930

-920

-910

-900

-890

-880

GGAQc

Ecut(ev)0 200 400 600 800 1000 1200 1400

Tota

l ener

gy

(ev)

-3680

-3660

-3640

-3620

-3600

-3580

-3560

-3540

-3520

-3500

-3480

GGA

-3728.1091-3728.1420Convectional cell

-922.0743-922.0355Unit cell(ev)

Ecut=1200evEcut=900evEcut(ev)

1

Total energy versus cut off energy with Kerker method and GGA approximation

Total energy versus cut off energy obtained by Kerker methodunit cell conventional cell

-3726.7754-3726.7590Convectional cell(ev)

-928.1509-928.165250unit cell energy(ev)

Ecut=1300�Ecut=1200cut of energy (ev)

GGA

Ecut(ev)200 400 600 800 1000 1200 1400

Tot

al E

ner

gy(

ev)

-925

-920

-915

-910

-905

-900

-895

-890

-885

Qc

Ecut(ev)0 200 400 600 800 1000 1200 1400

Tota

l ener

gy

(ev)

-3750

-3700

-3650

-3600

-3550

-3500

-3450

-3400

GGA

1

1

Comparison of total energy of four different configurations of cubic ice phases

-3745.733617-3745.74040-3745.74815-3745.761838Energy (ev)

(by other)

-3685.9369-3685.9413-3685.9682-3686.7968Calculated energy(ev)

4 3 2 1Configuration

1

Total energy (ev) for unit cell and conventional cell:

-3750.03-3741.032-935.258Total energy by Kerker [Ecut=1200]

-3686.81-3676.2-919.050Total energy by Qc-method [Ecut=900]

Convectional cell energy

Unit cell energy*4Unit cell energy

Cut of energy

1

The total energy versus volume for unit cell calculation and Qc-tuning method

V(A0)3

20 40 60 80 100 120

To

tal e

ner

gy

-920

-918

-916

-914

-912

-910

-908

GGA

LDA

1

Parameters obtained by Qc-tuning for unit cell without motion of Ion

Parameters obtained by Qc-tuning for unit cell with motion of Ion

5.1286.68476274.6789-914.5774.82010.1291GGA

9.626.970484.6671-908.6274.25670.08614LDA

errora(Å) Vo Eo Bop Bo approach

0.226.336762.6117-919.0501.03980.2152GGA

0.226.226762.6117-912.2061.01020.2208LDA

errora(A)VoEoBopBoapproach

1

The effect of D and P orbitals of hydrogen atom on structure parameterStructural parameters obtained by using unite cell and d orbital of hydrogen atom in

pseudo potential in Qc-tuning method…

Total energy versus volume for unite cell using d orbital of H atom and Qc-tuning method

0.336.336763.6117-912.2901.52320.21256LDA

0.336.336762.6117-918.9890.9935480.2261GGA

errora(A)VoEoBopBoapproach

V(A0)3

20 40 60 80 100 120

To

tal e

ner

gy

-920

-918

-916

-914

-912

-910

-908

GGA

LDA

1

Effect of various of Rc in structural properties of cubic ice

Variation of structural parameters versus cut off radius for GGA approximation and Qc-tuning method in calculation of unit cell

0.336.336763.6117-919.0010.65170.3189Rc(s,p)H=1.228(a.u)

Rc(d)H=1.25(a.u)

Rc(s,p,d)O=1.8(a.u)

0.336.33763.6117-918.1321.392050.271728 RcH=1.15(a.u)

RcO=1.8(a.u)

19.597.6040109.921-927.2192.688760.10228Rc(s,p,d)H=1.228(a.u)

Rc(s,p,d)O=1.5(a.u)

errora (Å) Vo Eo Bop Bo

1

Length of O-H band 0.97 Angstrom 0.92 Angstrom 0.95 Angstrom

The curves of total energy versus volume of conventional cell computed from pseudo potential obtained by Qc-tuning method

GGA

V(Ao)3

100 150 200 250 300 350 400 450 500

To

tal E

nerg

y(ev

)

-3690

-3680

-3670

-3660

-3650

-3640

GGA

LDA

GGA

V(Ao)3

100 150 200 250 300 350 400 450 500

Tot

al E

ner

gy(

ev)

-3690

-3685

-3680

-3675

-3670

-3665

-3660

-3655

-3650

-3645

GGA

LDA

GGA

V(Ao)3

100 150 200 250 300 350 400 450 500

To

tal E

ner

gy(

ev)

-3690

-3685

-3680

-3675

-3670

-3665

-3660

-3655

-3650

-3645

GGA

LDA

1

Calculated length of O-H band for cubic ice Different values for it has been reported

The length of O-H band is 0.97 Å

6.5190 277.047-3685.932.98810.16650.93

6.4107 263.466-3686.622.287050.2036540.95

6.3562256.798-3686.812.121720.2224810.97

a(Å) Vo Eo Bop Bo O-H(Å)

1

The Band Structure of cubic ice obtained by Wien2k soft wear. The gap energy is 5.4 ev

1

Conclusion:A)- We find the right configuration for Cubic Ice. B)-To compute exchange correlation energy, GGA and LDA approximations reach to the same value, but cell parameter which obtained by GGA method is slightly closer to the experimental value. C)-In pseudo potential calculation there will be no different in the outcome results if we consider p or d for exited state of hydrogen atom.D)- Structural determination for unit cell and conventional cell gives the same results which indicated that the positions of atoms have been calculated correctly. E)- We deduce the length of O-H band in cubic Ice which is 0.97 angstrom.

1

1

Minimizing the functional energy:

1- Solution to single equations of Kohn-Shem for single particle2- Conjugate gradient procedure

)()(21

)()()()(2

οοοο xxxx

FxxxFxxxFxF

ji

T −∂∂

∂−+∇−+=

1

.

:

Vxc

:

[ ]{ [ ]})()()()( rnVrnVrVrV xcHps

l

Unscreenpsl

���� +−=

))(())(())()(( rnVrnVrnrnV valHcoreHvalcoreH

���� +=+

[ ]{ [ ]})()()()( rnVrnVrVrV valxcvalHps

l

Unscreenpsl

���� +−=

1

rc .

1

)(...)()(

...

)(...)()(

)(...)()(

!

1),...,,(

21

22212

11111

321

rrr

rrr

rrr

Nrrr

NNN

N

N

���

���

���

���

ψψψ

ψψψ

ψψψ

ψ =

)()()( σχϕψ iiiii rr�� =

One of the problems in Hearty approximation is that with changing the positions of electrons in the system,the total wave function is not

ant symmetry,and the Pouli rule is not valid.To solve this problem fock:

1

(Ab initio)

.1-

BHS

.2-

1

For structural determination we have to put pressure on the crystal. We increase and decrease the cell parameter and using Mornagon equation of state.

cteBVV

B

BVvE

B

++−′

=

′

]11

)([][

ο

ο ο