CHEM 834: Computational Chemistry

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CHEM 834 Computational Chemistry
Quantum Chemical Methods 1
March 10, 2008
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Topics
last time

• overview of computational chemistry

• potential energy surfaces

today

• quantum chemical methods

• review of quantum mechanics

• variational principle

• molecular orbitals

• Slater determinant wavefunctions

• Hartree-Fock calculations

• Hartree-Fock energy

• Fock operator

• self-consistent solution of the Fock operator

• availability in Gaussian

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Quantum Chemical Methods
quantum chemical methods employ a quantum
mechanical treatment of the electrons without
resorting to parameterization
also called

• ab initio

• electronic structure methods

• first principles

• molecular orbital calculations

What do we get from these calculations?

• electronic wavefunction ? contains all
  information regarding the system

• can calculate electronic energy

• useful for exploring the potential energy surface

• needed to calculate reaction energies and barriers

• can describe changes in electronic structure/bonds

• can study changes in molecular orbitals

• can calculate other properties related to the
  electronic structure

• various spectra

• can approximate quantities like atomic charges
  and bond orders

• reactivity indices

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Quantum Mechanics Review
Classical Mechanics (Newton et al.)

• objects are treated as particles with
  well-defined positions and momenta

• behaviour is governed by Newtons equations (or
  equivalent)

F ma

• appropriate for objects ranging from atoms to
  planets

Quantum Mechanics (Heisenberg, Schrödinger, et
al.)

• objects are described as probability distributions

• cannot know exactly a particles position and
  momentum simultaneously

? Heisenberg uncertainty principle

• all information regarding the system is contained
  in the wavefunction

? wavefunction obtained through Schrödinger
equation (or equivalent)

• needed to describe behaviour of small, light
  particles (electrons, maybe protons)

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Postulates of Quantum Mechanics

• the state of a system is described by a function
  ?(r,t) called a wavefunction

Postulate 1

• ?(r,t) is single-valued, continuous,
  quadratically integrable

• the probability that the particle is in volume d?
  at position r and time t is

• since the particle must be somewhere all the time

d? indicates integration over all coordinates
indicates the complex conjugate of ?

• i ? -i, e.g.

bold fonts indicate vectors
e.g. r (x,y,z)
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Postulates of Quantum Mechanics

• the state of a system is described by a function
  ?(r,t) called a wavefunction

Postulate 1

• ?(r,t) is single-valued, continuous,
  quadratically integrable

• the probability that the particle is in volume d?
  at position r and time t is

• since the particle must be somewhere all the time

Postulate 2

• every physical observable is represented by a
  linear Hermitian operator

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Linear Hermitian Operators
Operator

• a rule that turns a function into another function

• designated with a hat (sometimes Ill use
  italics)

Linear Operator

• an operator with the following properties

Hermitian Operator

• an operator with the following property

• guarantees real values for associated property

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Postulates of Quantum Mechanics
Postulate 2

• every physical observable is represented by a
  linear Hermitian operator

• quantum mechanical operators are analogues of
  classical expressions using

• for example, consider the kinetic energy

• classically

• quantum mechanically

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Postulates of Quantum Mechanics

• in any measurement of the observable associated
  with operator B the only values that will ever be
  observed are eigenvalues bi that satisfy

Postulate 3
eigenfunction of B
eigenvalue of B

• also called eigenstate

• ?i is an eigenfunction of operator B if operating
  on ?i with B gives back ?i multiplied by a
  constant

• constant is called an eigenvalue, value of
  property B when system is in state i

example
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Eigenvalues and Eigenfunctions

• eigenfunctions of Hermitian operators are
  orthonormal

• basically says system is in state i or j, but not
  both

• lets us calculate the eigenvalue for a given state

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Postulates of Quantum Mechanics

• in any measurement of the observable associated
  with operator B the only values that will ever be
  observed are eigenvalues bi that satisfy

Postulate 3

• this postulate introduces notion of discretized
  states, which is at the heart of quantum mechanics

Postulate 4

• an arbitrary state ? can be represented as a
  linear combination of the eigenfunctions of a
  quantum mechanical operator

• called superposition principle

• system must adopt a specific state upon
  measurement

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Postulates of Quantum Mechanics

• the average value of property B at time t is

Postulate 5

• time development of a state is given by the
  time-dependent Schrödinger equation

Postulate 6

• H is the Hamiltonian total energy operator

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Time-Independent Schrödinger Equation
for most chemical problems V is independent of
time
usually Coulomb potential

• energy of state specified by ? is constant at all
  times

• ?(r) is the solution to the time-independent
  Schrödinger equation

time-independent Hamiltonian
time-independent wavefunction
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Summary of Quantum Mechanics
solving the time-independent Schrodinger equation
gives us the energy and wavefunction

• the wavefunction contains all the information
  that can be known for the system

• there is a different wavefunction for each state

• wavefunctions are orthonormal

• the square of the wavefunction is interpreted as
  a probability

• P(r1) is the probability of finding the electron
  in volume element dr1 at position r1

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Summary of Quantum Mechanics
we calculate properties by operating on
wavefunctions

• operators are analogues of classical expressions
  for properties

• the values of the properties are eigenvalues of
  the operators

• the wavefunction is an eigenfunction of the
  operator

• the value of a property for a specific state is

any wavefunction can be expressed as a linear
combination of eigenfunctions of any quantum
mechanical operator

• this mathematical property will be useful when we
  develop quantum chemical methods

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Molecular Hamiltonian
quantum chemical methods strive to solve the
time-independent Schrödinger equation
for a system of N electrons in the presence of M
nuclei using only mathematical approximations
? no parameterization/fitting to experimental
data
2 nuclei, 2 electrons
terms included in the Hamiltonian
1 electron-electron interaction
1 nucleus-nucleus interaction
4 nucleus-electron interactions
kinetic energy of electrons and nuclei
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Molecular Hamiltonian
2 nuclei, 2 electrons
terms included in the Hamiltonian
1 electron-electron interaction
1 nucleus-nucleus interaction
4 nucleus-electron interactions
kinetic energy of electrons and nuclei
simplifications
1. Born-Oppenheimer approximation
2. atomic units
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Born-Oppenheimer Approximation

• nuclei are much heavier than electrons

? mproton 2000 melectron
? nuclei move much more slowly
? consider electrons as moving in field of fixed
nuclei

• implications

? nuclear kinetic energy can neglected
? nuclear-nuclear repulsion energy is constant
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Born-Oppenheimer Approximation

• solution to Schrodinger equation for electronic
  Hamiltonian

• is the electronic wavefunction

• depends explicitly on electronic coordinates, ri,
  and parametrically on the nuclear coordinates, RI

• total energy also depends parametrically on RI

• relationship between Etot and RI yields the
  potential energy surface

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Atomic Units
simplify by changing units
distance a0 bohr radius 5.2918 x 10-11 m
energy Hartree 4.3598 x 10-18 J 627.51
kcal/mol
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Quantum Chemical Methods
we want to calculate the total molecular energy
using
where Htot is
since the last term is a constant, we really need
to solve
where Helec is

• will give us the electronic energy

• will give us the electronic wavefunction ? used
  to get other properties

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Quantum Chemical Methods
it is not possible to solve the Schrodinger
equation exactly for a many-body system
electron-electron terms are the problem
called the many-body problem

• the motion of electron 1 affects the motion of
  electron 2, and vice-versa

• prevents separability of the Hamiltonian

• means we cant break down the Hamiltonian into
  independent equations for electrons 1 and 2

• many-body problem is not unique to quantum
  mechanics and electrons

• the sun-moon-earth system cannot be described
  rigorously with classical mechanics because of
  many-body effects

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Quantum Chemical Methods
it is not possible to solve the Schrodinger
equation exactly for a many-body system
electron-electron terms are the problem
Strategies used in quantum chemistry
1. treat electron-electron interactions as a
minor perturbation

• called perturbation theory

• more on this later

2. guess a suitable form of ? and optimize it

• called variational method

• basis for most of quantum chemistry

• well look at this now

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Variational Method
Example for the Hydrogen atom
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Variational Method
Example for the Hydrogen atom
we want to pick ? to give the best agreement with
the real solution
Obvious questions
1. can we really do this?
2. how do we optimize ? (i.e. how do we pick
?)?
3. what form of ? should we use?
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Variational Principle Some Background
there exists an infinite set of wavefunctions,
??, such that
where the energies E? are the eigenvalues for a
particular state
since the wavefunctions are orthonormal
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Variational Principle Some Background
energy of a specific state

• we cant solve this because we dont know what ?j
  is

variational approach (basic idea)

• lets take a guess at the form of ?j and try to
  get as close as possible to the real wavefunction

• trial wavefunction must meet certain constraints
  to be valid

• more on this later

4th postulate of quantum mechanics

• an arbitrary wavefunction can be expressed as a
  linear combination of the set ?i

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Variational Principle Some Background
trial wavefunction

• choose ? to be orthonormal

• dont know exact form of ?i

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Variational Principle Some Background
trial wavefunction

• choose ? to be orthonormal

• dont know exact form of ?i

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Variational Principle Some Background
what we know
compare the trial energy with the ground state
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Variational Principle Some Background
what we know
compare the trial energy with the ground state
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Variational Principle Some Background
what we know
compare the trial energy with the ground state
and with some rearrangement

• this is called the variational integral

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Variational Principle
energy of trial function, ?
true ground state energy of the system
normalization factor
for any suitable trial wavefunction

• the energy of the trial function is guaranteed to
  be higher than the real ground state energy

• if we happen upon the real wavefunction we get E0

• otherwise, we get an upper bound on E0

• basic principle

trial wavefunction with lower energy is a
better trial wavefunction
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Construction and Optimization of Trial
Wavefunctions
nice concept, but how do we use it?
Linear variation

• pick a trial function that is a linear
  combination of basis functions

well-defined functions
coefficients in linear expansion

• modify coefficients ai to minimize the energy

the best trial wavefunction gives the lowest
energy
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Construction and Optimization of Trial
Wavefunctions
trial function is a linear combination of basis
functions
Back to the Hydrogen atom
a single gaussian function is not a good trial
wavefunction!!!
a11.0, ?10.5
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Construction and Optimization of Trial
Wavefunctions
trial function is a linear combination of basis
functions
Back to the Hydrogen atom
a linear combination of gaussian functions is a
good trial wavefunction!!!
a10.1, ?13.5
a20.5, ?20.6
a30.4, ?30.2
optimized to minimize energy
fixed to define the basis function
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Linear Variation
the best trial wavefunction minimizes the energy
to optimize ?, we need a relationship between E
and ai
in compact notation
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Linear Variation

• energy of the trial wavefunction

• we want to minimize this

? lower energy better wavefunction

• Hij and Sij are constants

• ais are variables

Minimization

• minima are stationary points

this is like a geometry optimization on the
potential energy surface, but instead of
coordinates, we have coefficients
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Secular Equations
Working out the derivatives

• N secular equations

• N unknowns (ais)

• ais define ? with energy E

• to solve, we need E first

• get E from secular determinant

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Secular Determinant
Secular determinant

• polynomial of order N with N roots

• analogous to how ax2 bx c 0 gives 2 roots

• roots correspond to energies

• E0 ? E1 ? E2 ? E3 ? E4 ? EN-1

• can be shown that each Ej is an upper bound on
  the real energy of state j

• the Ejs can be used to get a set of ais for
  each state by solving the secular equations

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Secular Equations
we can get N trial wavefunctions from N basis
functions
E0, E1, E2, ,EN-1
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Linear Variation - Summary
Procedure
2. evaluate Hij and Sij
3. construct secular determinant and solve for
Ej
4. solve
for each Ej
Results
set of coefficients ai for each state j that
give lowest energy for a specific trial
wavefunction
Note

• more basis functions, ?i, gives more coefficients
  to tweak

• leads to a lower energy, and better wavefunction

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Variational Method
Example for the Hydrogen atom
we want to pick ? to give the best agreement with
the real solution
Obvious questions
1. can we really do this? ? variational principle
2. how do we optimize ? (i.e. how do we pick
?)? ? linear variation method
3. what form of ? should we use?
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Trial Wavefunctions
how do we select a trial wavefunction for
molecules?
1. Assumptions

• each electron occupies a molecular orbital
  (analogous to atomic orbitals)

• molecular orbitals are orthonormal

2. Constraints

• trial wavefunction must satisfy Pauli exclusion
  principle

• trial wavefunction must be antisymmetric

• trial wavefunction must not distinguish between
  electrons

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Molecular Orbitals
in atoms

• each electron occupies 1 atomic orbital

e.g. 1s, 2s, 2pz,

• atomic orbitals are centered on nucleus

• atomic orbitals are orthonormal

in molecules

• each electron occupies 1 molecular orbital

• molecular orbitals can be delocalized across all
  atoms

• molecular orbitals are orthonormal

C-C ?-bond in ethene
C-C ?-bond in ethene
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Molecular Orbitals
molecular orbitals depend on spatial coordinates
and spin
Spatial orbital, ?i(r)

• function of the position vector, r

• describes the spatial distribution of the
  electron

• form an orthonormal set

Spin functions

• Helec does not include spin ? would need a
  relativistic treatment to get spin

• artificially introduce orthonormal spin
  functions g(?) ?(?) or ?(?)

• ? represent spin coordinates

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Molecular Orbitals
molecular orbitals depend on spatial coordinates
and spin
Spin orbital, ?i(x)

• spin orbitals are used in quantum chemical
  calculations

• treatment of spin increases the number of degrees
  of freedom

spin coordinates
spatial coordinates

• for a system with N electrons, there are 4N
  degrees of freedom

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Constraints on the Trial Wavefunction
for variational principle to apply, the trial
wavefunction must satisfy the same constraints as
the real wavefunction
1. Pauli exclusion principle

• for atoms ? no two electrons can have same set of
  quantum numbers

• for molecules ? no two electrons can be in same
  state

? no two electrons can have same xi
2. Antisymmetry

• ? must be antisymmetric with respect to
  interchange of coordinates x between two electrons

• if ? is antisymmetric, the Pauli exclusion
  principle will be satisfied

3. Indistinguishability

• uncertainty principle ? we cant know exact
  positions and velocities of electrons

• ? must not distinguish between particles

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Molecular Orbitals ? Trial Wavefunction
we want the form of the trial wavefunction to
closely resemble the true wavefunction
cannot treat electron-electron terms exactly
lets forget about e-e interactions for now
solve Schrödinger equation
solution is product of molecular orbitals
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Hartree Product

• called a Hartree product

• appealing because it is simple from a
  mathematical standpoint

Hartree product is an independent-electron
wavefunction

• probability that electron 1 is in dx1 is
  independent of probability that electron 2 is in
  dx2, etc.

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Hartree Product
consider flipping 2 coins
probability of both coins landing heads up
probability of coin 1 being heads
probability of coin 2 being heads
if a probability can be written as a product of
probabilities of individual events, those events
are independent of each other
consider a 2 electron example
probability electron 1 is in dx1 at x1 and
electron 2 is in dx2 at x2
probability electron 1 is in dx1 at x1
probability electron 2 is in dx2 at x2
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Hartree Product

• called a Hartree product

• appealing because it is simple from a
  mathematical standpoint

Hartree product is an independent-electron
wavefunction

• probability that electron 1 is in dx1 is
  independent of probability that electron 2 is in
  dx2, etc.

• in reality the motion of electron 1 should depend
  on the motion of all the other electrons ?
  electron correlation (more later)

implications

• Hartree product neglects instantaneous
  electron-electron interactions

• for an N electron system, electron 1 feels the
  average Coulomb repulsion of the other N-1
  electrons

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Hartree Product
Coulomb energy
point charges at r1 and r2
Q1(r1)
Q2(r2)
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Hartree Product
Coulomb energy
Q2 induces a potential at r1
r1
Coulomb potential
Q2(r2)
Q3(r3)

• electrostatic potential at r1 due to a point
  charge at r2

Q5(r5)

• for multiple charges

Q4(r4)
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Hartree Product
electrons are not point charges, but instead are
charge distributions
consider a two-electron system

• charge in dr2 at r2 from the electron in ?2

r1

• the potential at point r1 from the charge in dr2
  at r2 from the electron in ?2 is

charge cloud of electron in ?2

• but the electron is spread over all values of r2

• for continuous values of r2, the sum becomes an
  integral

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Hartree Product
bring in the second electron

• charge in dr1 at r1 from the electron in ?1

• the Coulomb repulsion the electron in ?1
  experiences in dr1 at r1 is

charge cloud of electrons in ?1 and ?2

• integrating over all values of r1 gives the total
  electron-electron Coulomb energy

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Hartree Product
for an N electron system

• the Coulomb potential the electron in ?i feels in
  dr1 at r1 is

• this is simply the electrostatic repulsion from
  the other N-1 electrons in the system

• since the electron distributions are independent,
  this represents the average electrostatic
  potential ? instantaneous electron-electron
  repulsion is not considered

Hartree product wavefunction treats each electron
as moving in the average field of the other N-1
electrons in the system!!!

• significant limitation because electrons do
  interact instantaneously

• but, not a surprising result because the Hartree
  product is the solution of the Hamiltonian
  without any electron-electron interactions

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Hartree Product
lets test ?HP against the criteria for a valid
wavefunction
1. Indistinguishabilty

• distinguishes between electrons

• invalid form of wavefunction

2. Antisymmetry

• not antisymmetric

• invalid form of wavefunction

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Slater Determinant Wavefunction
deficiencies of ?HP can be overcome by taking
linear combinations of product wavefunctions
1. Indistinguishabilty

• does not distinguish between electrons

• electron 1 can be found in either orbital

• electron 2 can be found in either orbital

• valid form of wavefunction

2. Antisymmetry

• antisymmetric

• valid form of wavefunction

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Slater Determinant Wavefunction
?12 is a Slater determinant
normalization constant
determinant
Determinants

• square array of elements

• lines on left and right (dont confuse with
  matrices that use brackets)

• functions as a shorthand for writing the sum of
  products

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Slater Determinant Wavefunction
Properties of Determinants (D)
1. if all elements in a column or row are 0, D 0
2. multiplying a row or column by k multiplies D
by k
3. switching two columns or two rows changes the
sign of D
4. if two columns or two rows are identical, D 0
5. if two columns or two rows are multiples of
each other, D 0
6. multiplying a column (or row) by k and adding
it to another column (or row) leaves D unchanged
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Slater Determinant Wavefunction
virtually all quantum chemical methods use Slater
determinants
For an N electron system

• columns labeled by molecular orbitals

• rows labeled by electrons

• shorthand notation

Based on properties of determinants

• switching two rows changes sign of ?SD ?
  antisymmetry

• if two columns (molecular orbitals) are identical
  ?SD 0 ? two electrons cant occupy same orbital
  (Pauli exclusion principle)

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Slater Determinant Wavefunction
Hartree product was an independent electron
wavefunction
what about Slater determinants?
consider two electron system with electrons of
opposite spin
square to get probability
define P(r1,r2)dr1dr2 probability of finding
electron 1 in dr1 and electron 2 in dr2

• probability is just products ? electrons of
  opposite spin are independent with Slater
  determinant

• in fact, P(r1,r1) ? 0 ? with Slater determinant
  two electrons of opposite spin can occupy the
  same point in space

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Slater Determinant Wavefunction
Hartree product was an independent electron
wavefunction
what about Slater determinants?
consider two electron system with electrons of
same spin
using analysis on preceding slide

• probability is not just products ? electrons of
  same spin are correlated with Slater determinant

• called exchange correlation because last two
  terms exchange electron coordinates between
  different spatial orbitals

• consequence of antisymmetrizing the wavefunction,
  and related to Pauli exclusion principle

• P(r1,r1) 0 ? with Slater determinant two
  electrons of same spin cannot occupy the same
  point in space

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Slater Determinant Wavefunction
For an N electron system

• columns labeled by molecular orbitals

• rows labeled by electrons

• shorthand notation

Physics captured

• electrons of opposite spin are treated
  independently

? neglects instantaneous Coulomb interactions

• electrons of opposite spin are correlated through
  exchange interactions

? accounts for Pauli exclusion principle
? significant improvement over Hartree product
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Hartree-Fock Calculations
Hartree-Fock calculations are the basis for
virtually all quantum chemical methods

• ab initio methods are built upon Hartree-Fock
  calculations

• in practice, density functional theory
  calculations look very much like Hartree-Fock
  calculations

• semi-empirical molecular orbital methods involve
  approximations with the framework of Hartree-Fock
  theory

Other common names for Hartree-Fock calculations

• self-consistent field (SCF) calculations

• these names are technically incorrect, but you
  may see them in the literature

• molecular orbital (MO) calculations

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Hartree-Fock Calculations
The basic idea of Hartree-Fock calculations
1. the full Hamiltonian within the
Born-Oppenheimer approximation
2. a trial wavefunction consisting of one Slater
determinant
3. molecular orbitals expressed as linear
combinations of basis functions
basis function with a fixed form
artificial spin function
coefficient in linear expansion (called molecular
orbital coefficients)
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Hartree-Fock Calculations
The basic idea of Hartree-Fock calculations
1. the full Hamiltonian within the
Born-Oppenheimer approximation
2. a trial wavefunction consisting of one Slater
determinant
3. molecular orbitals expressed as linear
combinations of basis functions
4. variational optimization of ? using the
molecular orbital coefficients as variational
parameters
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Linear Combination of Basis Functions
Hartree-Fock molecular orbitals are expressed as
linear combinations of basis functions
Basic motivation

• linear combination of atomic orbitals

• consider H2

• molecular orbitals are expressed as linear
  combinations of atomic orbitals

• K atomic orbitals yields K molecular orbitals

H
H

• N molecular orbitals with lowest energy are
  occupied and all others are called unoccupied

1sL
1sR

• applies to all molecules

70
Linear Combination of Basis Functions
Hartree-Fock molecular orbitals are expressed as
linear combinations of basis functions
basis functions other than atomic orbitals

• linear combination of atomic orbitals

• basis functions dont have to be atomic orbitals

• we can choose any suitable well-defined
  mathematical function

• well discuss basis functions more in a later
  lecture

atomic orbital basis set
alternative basis set

• chemically intuitive

• still mathematically valid

C
H
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Hartree-Fock Energy
to use the variational method, we need a
relationship between the energy and the MO
coefficients
for any normalized wavefunction
in Hartree-Fock theory
and if you work it out, the energy becomes
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Hartree-Fock Energy
to use the variational method, we need a
relationship between the energy and the MO
coefficients
recall that molecular orbitals are linear
combinations of basis functions
so, now we have a direct relationship between the
Hartree-Fock energy and molecular orbital
coefficients

• later, well look at how to variationally
  optimize the MO coefficients

• now, lets look at the energy expression itself

73
Hartree-Fock Energy
in terms of the MOs, the Hartree-Fock energy is
or in standard short-hand notation
exchange integral
Coulomb integral
one electron energy
two electron energy
74
One Electron Energy
same molecular orbital
details

• involves only 1 molecular orbital ? one-electron
  energy

75
One Electron Energy
kinetic energy of electron in ?a
Coulombic attraction between M nuclei and
electron in ?a
details

• involves only 1 molecular orbital ? one-electron
  energy

• electronic kinetic energy

• nuclear-electron Coulombic attraction

• must sum over all electrons in the system

76
Coulomb Integral
two different molecular orbitals
details

• involves 2 molecular orbitals

• called a two-electron integral

• have to sum over all pairs of electrons to get
  total energy

77
Coulomb Integral
probability of finding electron a in dx1
details

• involves 2 molecular orbitals

• called a two-electron integral

• have to sum over all pairs of electrons to get
  total energy

• represents Coulomb interaction between two charge
  clouds representing the electrons in molecular
  orbitals a and b

• probability of finding electron a in dx1 is
  independent of finding electron b in dx2

• accounts for electrostatic energy resulting from
  electron a moving in average field of electron b

• Jab is non-zero even when x1 x2

• electrons can exist at same point in space

• violates Pauli exclusion principle for electrons
  of same spin

78
Exchange Integral
two different molecular orbitals
details

• involves 2 molecular orbitals

• called a two-electron integral

• have to sum over all pairs of electrons to get
  total energy

• exchanges coordinates of electrons in molecular
  orbitals a and b

• does not have a simple classical analogue like
  Coulomb integral

• accounts for exchange energy (Pauli repulsion
  between electrons)

• consequence of using antisymmetric wavefunction

• is only non-zero for electrons of the same spin

• Pauli repulsion doesnt affect electrons of
  opposite spins

• cancels out Coulomb energy arising from two
  electrons with the same spin being at the same
  point in space

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Two Electron Energy
sum over all pairs of electrons

• prefactor of ½ prevents double-counting of
  interactions

Jab has a positive sign

• Coulomb interaction between electrons is repulsive

Kab has a negative sign

• exchange interactions make electrons of the same
  spin avoid each other

• Jab does not account for this correlation,
  overestimates repulsive energy

• -Kab removes this overestimation by reducing the
  energy

Jaa Kaa

• Jaa is the interaction of an electron with itself

• Kaa cancels out exactly this spurious interaction

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Hartree-Fock Energy
Hartree-Fock energy captures

• kinetic energy of electrons

• nuclear-electron attraction

• electron-electron Coulomb repulsion

• this interaction is treated in an average sense

• instantaneous electron-electron Coulomb
  interactions are not considered

• exchange interactions

• accounts for Pauli repulsion/exclusion principle

• electrons of the same spin avoid each other ?
  decrease in Coulomb repulsion

All of these quantities are approximate because
the wavefunction has an approximate form!!!!
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