easy slides to understand the hartree method in basic physical chemistry of BS_6th semester in Pakistani universities.

1. Topic: HATREE METHOD . 1
Prepared By group no. 2
Msc-chemistry_ 2A

2. SETTING UP THE PROBLEM
 A molecule is a collection of charged particles. Positive nuclei and negative electrons
bound together to form a stable system.
 The interaction between charged particles is described by coulomb law.
= V( ) = =
2

3. The Schodinger Equation
 Kinetic energy of nuclei
 Kinetic energy of electrons
 Coulombic energy between nuclei
 Coulombic energy between electrons
 Coulombic energy between nuclei and
electrons
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==
=
=
=

4. Born oppenheimer approximation
 Nuclei are much heavier than electrons(ma>me ≥1836)and moves much slower.
 Effectively electrons adjust themselves instantaneously to nuclear configurations.
 Electrons and nuclear motions are uncoupled ,thus the energies of the two are separable.
 For a given nuclear configuration ,one calculates the electronic energy.
 As nuclei moves continuously , the points of electronic energy joint to form a potential energy surface
on which nuclei move 9(the potential energy surface).
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5. Many electron wave function
▰ .
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PAULI PRINCIPLE ; Two electrons can not have all quantum number equal .
Hartree product ; all electrons are independent ,each in its own orbital.
This requires that total wave function is anti symmetric whenever one exchanges two electrons
coordinates .
( )
=

6. A SLATER DETERMINENT
satisfies the Pauli's principle
▰
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……….. ………

7. SOLUTION:
Use the variational principle to generate approximate solutions.
In Practice: Generate the “best” trial function that has a number of
adjustable parameters. The energy is minimized as a function of these
parameters.
In our case: minimize the energy with respect to the orbitals .
Self-consistent Field ( SCF ) Theory
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GOAL: Solve the electronic schrodinger equcation, HeѰ=EѰ
PROBLEM: Exact solutions can only be found for one-electron system e.g. H2
+ .

8.  Each electron feels all the other electrons as a whole (field of charge). i.e, an
electron moves in a mean-field generated by all the other electrons .
 A fock operator F is introduced for a given electron in the i-th orbitals :
FiØi = ԑiØi
Fi = kinetic energy terms potential energy term Averaged Potential
energy of the given electron + due to fixed nuclei + term due to other
electrons .
Øi is the i-th molecular orbital, and ԑi is the correspending orbital energy.
Hartree-Fock Approximation
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9. The Fock Operator
Core Hamiltonian
operator
Kinetic energy term
and nuclear attraction
for the given electron.
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=
Coulombic operator
Coulombic energy term
for the given electron due to
an other electron.
Exchange Operator
Exchange energy due to another
Electron(A pure quantum mechanical
term due to pouli’s Principle , no
classical interpretation).

10. Self-consistency
 Each electrons feels all other electrons as a whole(field of charge) i.e an electron
move in a mean-field generated by all other electrons.
 The fock operator for an electron in the i-th orbital contains information's of all the
other electrons ( in an averaged fashion), i.e. the fock equations for all electrons are
coupled with each other.
 All equation must be solved together (iteratively until self-consistency is obtained).
 _____Self-consistent field (SCF )method.
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11. Linear Combination of Atomic Orbitals
 Each one electrons molecular orbital's is approximated by a linear
combinations of atomic orbital's (Basis function).
 Ø= c1X1+c2X2+c3X3+…….
 Where Ø is the molecular orbital's wave function, Xi represents Atomic
orbit wave function, and Ci is the corresponding expansion coefficients.
 The resulting fock equation are call Roothaan-Hall equations .
 This reduces the problem of finding the best functional form for the
molecular orbital's to the much simpler one of optimizing a set of
coefficients(cn).
 One can vary the coefficients to minimize the calculated energy.
 At the energy minimum , and one has the best approximation of the true
energy____ that is what we want.
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12. Restricted Hartree-Fock method(RHF)
 A variation on the HF procedure is the way that orbitals are constructed to
show paired or unpaired electrons.
 If the molecule has a singlet spin, then the same orbital spatial function can
be used for both the a and b spin electrons in each pair. This is called the
restricted Hartree-Fock method (RHF).
 There are two techniques for constructing HF wave functions of molecules
with unpaired electrons. One technique is to use two completely separate
sets of orbitals for the a and b electrons.
 This is called an unrestricted Hartree-Fock wave function (UHF).
 This means that paired electrons will not have the same spatial distribution.
This introduces an error into the calculation, called spin contamination. Spin
contamination might introduce an insignificant error or the error could be
large enough to make the results unusable depending on the chemical system
involved. 12

13. Electron Correlation – Configuration Interaction
 Because electrons are negatively charged, electrons like to avoid each other.
When one electron moves, the position of other electron, the position of other
electron in the molecule adjust to minimize electron-electron repulsion. Therefor
the motion of electron are not independent, the motion of electron is correlated.
The current best method for adjusting molecular wave function for electron
correlation is called configuration interaction CI. CI Calculation allows the ground
state of molecule to mix with the excited state of molecule. For example molecule
of H2 , the ground state configuration and several excited states are diagrammed.
below
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HHHH HH
Ground excitation Single Excitation Double excitation
E E E

14. Moller-Plesset Pertubation Theory
 Correlation can be added as a perturbation from the Hartree-Fock wave
function.
 This is called Moller-Plesset perturbation theory.
 HF becomes a first-order perturbation.
 Thus, a minimal amount of correlation is added by using the second order
MP2 method.
 Third-order (MP3).
 Fourth-order (MP4).
 MP5 and higher calculations are seldom done due to the high computational
cost (N10 time complexity or worse).
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15.  Moller-Plesset calculations are not variational.
 There is also a local MP2 (LMP2) method.
 LMP2 calculations require less CPU time than.
 MP2 calculations. LMP2 is also less susceptible to basis set superposition error.
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16. An easy approach is to use perturbation theory
 The MP2 method uses second method perturbation theory to calculate the
mixing of various states. Remember that to calculate the correction of wave
function caused by a perturbation given as:
That is the new wave function is the combination of all the other wave function
k, but the wave function that are closest in energy to i are most important. In MP2
calculation, single excitation don’t contribute, only double or higher excitation
contribute. Eq. 1 is the MP2 result, of course for H2 only single and double
excitation allowed, since H2 has only 2 electron.
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Where

17.  UHF calculation are popular because they are easy to implement and run
fairly e½ciently.
 Another way of constructing wave functions for open-shell molecules is the
restricted open shell Hartree-Fock method (ROHF)
 In this method, the paired electrons share the same spatial orbital; thus, there
is no spin contamination.
 The ROHF technique is more difficult to implement than UHF and may require
slightly more CPU time to execute.
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18.  For singlet spin molecules at the equilibrium geometry, RHF and UHF wave
functions are almost always identical.
 RHF wave functions are used for singles because the calculation takes less
CPU time
 In a few rare cases, a singlet molecule has biracial resonance structures and
UHF will give a better description of the molecule (i.e., ozone).
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19. CONFIGURATION INTERACTION (CI)
 A configuration interaction wave function is a multiple-determinant wave function
 This is constructed by starting with the HF wave function and making new determinants by
promoting electrons from the occupied to unoccupied orbitals.
 Configuration interaction calculations can be very accurate, but the cost in CPU time is very high
 Configuration interaction calculations are classified by the number of excitations used to make
each determinant.

20. CONFIGURATION INTERACTION SINGLE-EXCITATION (CIS)
 If only one electron has been moved for each determinant, it is called a configuration interaction
single-excitation (CIS) calculation.
 CIS calculations give an approximation to the excited states of the molecule, but do not change
the ground-state energy.

21. SINGLE-AND DOUBLE EXCITATION (CISD)
Single-and double excitation (CISD) calculations yield a ground-state energy that has been
corrected for correlation.

22. TRIPLE-EXCITATION (CISDT) & QUADRUPLE-EXCITATION (CISDTQ)
Triple-excitation (CISDT) and quadruple-excitation (CISDTQ) calculations are done only when .
very-high-accuracy results are desired.

23. CONFIGURATION INTERACTION (CI)
 The configuration interaction calculation with all possible excitations is called a full CI.
 The full CI calculation using an infinitely large basis set will give an exact quantum mechanical
result.
 However, full CI calculations are very rarely done due to the immense amount of computer power
required.

24. LUMO+1:
LUMO:
HOMO:
HOMO-1:
CI SINGLES (CIS)
Excited State 1: Singlet-A" 4.8437 eV 255.97 nm
f=0.0002
14 -> 16 0.62380 3.0329 408.79
14 -> 17 0.30035 3.73
Excited State 2: Singlet-A' 7.6062 eV 163.01 nm
f=0.7397
15 -> 16 0.68354 6.0794 203.94
6.41
Excited State 3: Singlet-A" 9.1827 eV 135.02 nm
f=0.0004
11 -> 16 -0.15957 6.6993 185.07
12 -> 16 0.55680
14 -> 16 -0.19752
14 -> 17 0.29331
ACROLEIN EXAMPLE
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H
O

25.  Excited State 4: Singlet-A" 9.7329 eV 127.39 nm f=0.0007
 9 -> 17 0.19146
 10 -> 16 0.12993
 11 -> 16 0.56876
 12 -> 16 0.26026
 12 -> 17 -0.11839
 14->17 -0.12343
 Eigenvectors CIS/6-31G(d) and INDO/S
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