CHEM 834: Computational Chemistry

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CHEM 834 Computational Chemistry
Quantum Chemical Methods 5
March 24, 2009
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Topics
last time

• density functional theory

today

• electronic excited states

• basis sets

• atom-centered basis sets

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Electronic Excited States
the methods weve explored are designed to get
the ground state wavefunction or density and
energy what about excited states?
Electronic excitations

• electrons are promoted from occupied orbitals
  into higher energy states

• excited state has at least one electron in an
  orbital of higher energy than it could be (well
  focus on single excitations)

energy

• we are interested in calculating excitation
  energies (UV/Vis), transition probabilities,
  excited potential energy surfaces, circular
  dichroism, etc.

Methods

• configuration interaction singles (CIS)

• time-dependent DFT (TDDFT)

• CASSCF (we wont look at this, though)

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Configuration Interaction Singles
can we use the Hartree-Fock wavefunction to get
excited states?
In Hartree-Fock

• the Slater-determinant wavefunction represent the
  ground state for the system

• in Hartree-Fock theory, the occupied orbitals are
  optimized to minimize the energy, and represent
  the actual molecular orbitals in the molecule

virtual
energy

• in Hartree-Fock theory, the virtual orbitals do
  not affect the energy, and are not optimized

• so, simply moving an electron from an occupied to
  virtual orbital is not a good way to get an
  excited state wavefunction

occupied
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Configuration Interaction Singles
in the Hartree-Fock ground state
HF ground state virtual orbitals

• these are not optimized with respect to the energy

energy
HF ground state occupied orbitals

• these are optimized to minimize the ground state
  energy

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Configuration Interaction Singles
if we naively excite an electron without
reoptimizing the orbitals
HF ground state virtual orbitals

• these are not optimized with respect to the energy

• the singly-occupied orbital should be optimized

energy
HF ground state occupied orbitals

• these orbitals were optimized for the ground state

• they should be reoptimized because the electron
  distribution is different than in the ground state

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Configuration Interaction Singles
what we would like to do is optimize the
electronic structure while leaving one low
energy orbital unoccupied

• unfortunately, the standard minimization
  techniques will not let you minimize the orbitals
  while leaving one unoccupied

energy
Configuration Interaction Singles

• lets represent the single excited states as
  linear combinations of singly-excited Slater
  determinants

k of the excited state
?ia Slater determinant formed by exciting an
electron from occupied orbital i into virtual
orbital a
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Configuration Interaction Singles
excited states linear combinations of singly
excited Slater determinants

• by using HF Slater determinants, we avoid
  optimizing the molecular orbitals for the excited
  state

energy

• by including multiple excited Hartree-Fock Slater
  determinants, we compensate for the fact that the
  orbitals are not optimized for the excited state

• the procedure for getting the coefficients ciak
  is a bit complicated, but it is basically
  designed to ensure that ?k is orthogonal to the
  HF ground state and all other excited states

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Configuration Interaction Singles
what do the expansion coefficients mean?

• the coefficients indicate how much a particular
  singly-excited Slater determinant constributes to
  the excited state wavefunction

energy

• the magnitude of the coefficient matters

• this information can be used to determine which
  orbitals an electron is excited from and into for
  a particular excited state

• e.g. if c342 0.002 and c352 -0.8, the 2nd
  excited state primarily involves a transition
  between HF orbitals 3 and 5

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Time-Dependent DFT
DFT is formally a ground state method, but we can
get information regarding excited states with
time-dependent DFT
electronic excitations involve the interaction of
electrons with electromagnetic radiation
if a molecule is subject to a fluctuating linear
electric field
and the frequency-dependent polarizability is
which contains information about the excited
states and their energies
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Time-Dependent DFT
DFT is formally a ground state method, but we can
get information regarding excited states with
time-dependent DFT
taking ?0 as the Slater determinant form from the
Kohn-Sham orbitals, and employing Greens
function techniques we can solve for ?i and Ei
If you do this (we wont), you find
singly-excited Slater determinants based on the
ground state Slater determinant formed from
Kohn-Sham orbitals
TDDFT excited states
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Transition Probabilities
the energies of the ground and excited states let
us calculate the wavelengths of UV/Vis
absorptions how do we get the intensities?
the transition probability is directly related to
the intensity

• also called the oscillator strength

• evaluated with the transition dipole operator

• i and j can be any states

• usually i excited state, j ground state

• fij will equal zero for forbidden transitions

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Excited State Calculations in Gaussian
Gaussian can perform

• CIS and TDDFT

• gives excited state wavefunctions, excitation
  energies and transition probabilities

• can perform geometry optimizations on the excited
  states

• can calculate related quantities like rotational
  strengths for CD spectra

• can solve for any number of excited states

• can solve for excited states of singlet or
  triplet multiplicity

• activated by keywords CIS/basis or TD and
  functional/basis on the route line

• CIS/MP2

• adds MP2 corrections to CIS excited states

• Random Phase Approximation

• similar to TDDFT, but based on Hartree-Fock

• CASSCF/CASPT2

• multi-reference methods allow one to probe
  excited states (not for novices, and many
  experienced computational chemists)

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Excited State Calculations in Gaussview
To do a CIS calculation
change Ground State to CIS
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Excited State Calculations in Gaussview
To do a CIS calculation
select types of excitations to consider
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Excited State Calculations in Gaussview
To do a CIS calculation
pick number of excited states to solve
pick state of interest

• Gaussian will print out detailed analysis of this
  state

• geometry optimization will be performed on this
  state

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Excited State Calculations in Gaussview
To do a TDDFT calculation
change Ground State to TD-SCF
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Excited State Calculations in Gaussview
To do a TDDFT calculation
set method to DFT
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Excited State Calculations in Gaussview
To do a TDDFT calculation
select exchange-correlation functional
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Excited State Calculations in Gaussview
To do a TDDFT calculation
set type of excitations, number of states and
state of interest
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Example TDDFT Calculation of H2CO
This example

• TDDFT calculations of H2CO

• B3LYP functional

• 6-31G(d,p) basis set

• 3 lowest energy singlet excited states

• 1st excited state is state of interest

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Example TDDFT Calculation of H2CO
Input

• TDsingles specifies a time-dependent calculation
  of singlet excited states

• B3LYP is a DFT functional, indicating that this
  is a TDDFT calculation

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Example TDDFT Calculation of H2CO
Output
we are interested in finding out the excitation
energies, nature of the transitions, and the
transition probabilities
excited stated energy (eV) and wavelength (nm)
transition probability
c891, coefficient in linear expansion defining
excited state wavefunction

• this excitation primarily involves a transition
  between occupied orbital 8 and virtual orbital 9
  (the HOMO and LUMO)

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Molecular Orbitals
we represent our electronic structure with a set
of one-electron orbitals
in Hartree-Fock
in post-Hartree-Fock
C-C ?-bond in ethene
in DFT
C-C ?-bond in ethene
so, molecular orbitals (or Kohn-Sham orbitals)
are really important for quantum chemisty
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Linear Combination of Basis Functions
we represent molecular orbitals as linear
combinations of basis functions
basis function with a fixed form
artificial spin function
coefficient in linear expansion (called molecular
orbital coefficients)
what are the basis functions?

• basic functions can take on any mathematical form

• but the forms we choose should

1. provide a good representation of the electron
density

• electron density is large in core orbitals,
  bonds, lone pairs, etc.

• electron density is small far away from nuclei

2. allow us to perform computations easily
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Linear Combination of Basis Functions
we represent molecular orbitals as linear
combinations of basis functions
atomic orbital basis set
alternative basis set

• still mathematically valid

• chemically intuitive

C
H
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Linear Combination of Basis Functions
we represent molecular orbitals as linear
combinations of basis functions
atomic orbital basis set
alternative basis set

• still mathematically valid, too

• chemically intuitive

• but, extra basis functions probably wont
  contribute significantly

C
H
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Atom-Centered Basis Functions
most quantum chemical programs use basis
functions that are centered on specific atoms
position of nucleus on which ? is centered
We know

• wavefunction must exhibit cusp at the nucleus

• wavefunction must decay exponentially at large
  distances from nucleus

• these properties are satisfied by Slater
  functions

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Slater Orbital Basis Functions
Slater orbitals are a natural choice for
atom-centered basis functions
position of atom on which the function is centered
normalization constant
? controls the rate of decay
?s
r-RA bohr
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Slater Orbital Basis Functions
Advantages of Slater Orbitals
1. exhibit cusp condition
2. capture correct exponential decay
Disadvantages of Slater Orbitals
1. mathematically inconvenient

• cannot evaluate 3 and 4 center integrals
  analytically with Slater orbitals

• have to employ costly numerical methods

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3 and 4 Center Integrals
recall, in Hartree-Fock we have to evaluate
orbitals involving two molecular orbitals
if we expand the orbitals as linear combinations
of basis functions
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3 and 4 Center Integrals
if the basis functions are centered on specific
nuclei
B
A
we get a four-centered integral

• these cannot be evaluated analytically if the
  basis functions are Slater orbitals

D
C

• same is true of the four basis functions are
  centered on three different atoms

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Slater Orbital Basis Functions
Advantages of Slater Orbitals
1. exhibit cusp condition
2. capture correct exponential decay
Disadvantages of Slater Orbitals
1. mathematically inconvenient

• cannot evaluate 3 and 4 center integrals
  analytically with Slater orbitals

• have to employ costly numerical methods

In practice

• Slater orbitals are not used very often

• exceptions include

• ADF DFT code that uses Slater orbitals

• semi-empirical molecular orbital methods, which
  neglect three and four center integrals

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Gaussian Basis Functions
Gaussian-type basis functions offer a convenient
alternative to Slater-type functions
?g
r-RA bohr
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Slater versus Gaussian
how do Slater and Gaussian functions compare?
Gaussians

• decay too quickly

• do not exhibit a cusp at the nucleus

?

• but you can evaluate three and four center
  integrals analytically

r-RA bohr
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Contracted Gaussian Functions
can we take linear combinations of Gaussian basis
functions to make them look more like Slater
functions
coefficients
c1 0.154329 c2 0.535328 c3 0.444635
?
?1 2.22766 ?2 0.405771 ?3 0.109818
r-RA bohr
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Contracted Gaussian Functions
can we take linear combinations of Gaussian basis
functions to make them look more like Slater
functions
coefficients
do not capture cusp exactly
c1 0.154329 c2 0.535328 c3 0.444635
good agreement over a wide range of r-RA
?
?1 2.22766 ?2 0.405771 ?3 0.109818
r-RA bohr
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Contracted Gaussian Functions
can we take linear combinations of Gaussian basis
functions to make them look more like Slater
functions
degree of contraction
?d primitive exponent
primitive Gaussian function
contracted Gaussian 1 basis function
cd contraction coefficient
each contracted Gaussian is 1 basis function
the contraction coefficients and primitive
exponents are fixed in calculations
contraction coefficients are not varied in
calculations
molecular orbital coefficients are varied in
calculations
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Contracted Gaussian Functions
can we take linear combinations of Gaussian basis
functions to make them look more like Slater
functions
contraction coefficients
c1 0.154329 c2 0.535328 c3 0.444635
primitive exponents
?
?1 2.22766 ?2 0.405771 ?3 0.109818
degree of contraction 3
r-RA bohr
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Contracted Gaussian Functions
Advantages of Gaussian Orbitals
1. we can treat them mathematically
2. we can introduce nodal features
Disadvantages of Gaussian Orbitals
1. we have to use a lot of basis functions to
reproduce the behaviour of Slater orbitals
2. contraction coefficients and orbital
exponents have to be fit to either experimental
or other data tedious work without a clear
direction for improvement
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3 and 4 Center Integrals
if the basis functions are contracted Gaussians
centered on specific nuclei
we can evaluate these integrals, but there will
be a lot of them
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Nodal Behaviour
the radial components of atomic orbitals exhibit
nodes
by setting some contraction coefficients to be
negative, we can mimic nodal behaviour with
contracted Gaussians
2s orbital exhibits a node where wavefunction
changes sign
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Angular Behaviour
atomic orbitals exhibit angular dependence in
addition to radial dependence
real and directed 1s, 2p, 3d and 4f hydrogen-like
orbitals
http//winter.group.shef.ac.uk/orbitron/AOs/1s/ind
ex.html
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Angular Behaviour
we introduce angular behaviour by multiplying
Gaussian functions with angular functions
Spherical Harmonics
the most straightforward way to introduce the
correct angular behaviour is to use spherical
harmonics
l azimuthal quantum number (angular momentum)
l 0 ? s
l 1 ? p
l 2 ? d
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Angular Behaviour
we introduce angular behaviour by multiplying
Gaussian functions with angular functions
Spherical Harmonics
the most straightforward way to introduce the
correct angular behaviour is to use spherical
harmonics
m magnetic quantum number
m (-l, -l1,,l-1,l)
this approach will give 1s function, 3 p
functions, 5 d functions, etc. for each Gaussian
basis function
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Angular Behaviour
we introduce angular behaviour by multiplying
Gaussian functions with angular functions
Cartesian Gaussians

• spherical harmonics can be inconvenient to deal
  with

• instead, many codes use Cartesian Gaussian
  functions

i j k l

• for s orbitals, i j k 0

• for p orbitals, i 1, or j 1, or k 1

• this gives px, py, and pz orbitals

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Angular Behaviour
we introduce angular behaviour by multiplying
Gaussian functions with angular functions
Cartesian Gaussians

• spherical harmonics can be inconvenient to deal
  with

• instead, many codes use Cartesian Gaussian
  functions

• for d orbitals, i 2, j 2, k 2, i j 1, i
  k 1, or j k 1

• this gives 6 orbitals

• 3 correspond to dxy, dxz, and dyz

• linear combinations gives dz2 and dx2-y2

• the sixth one has s symmetry

• for f, g, h you get even more extra orbitals

• some programs can eliminate these

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Angular Behaviour
most programs can work with either spherical
harmonics or Cartesian angular functions
Spherical Harmonics

• use the minimum number of functions for each
  angular momentum

• facilitate easy interpretation of results, e.g.
  its easy to determine if s, p or d electrons are
  contributing to a particular molecular orbital

Cartesian Angular Functions

• mathematically convenient

• too many orbitals for higher angular momenta ?
  more computational effort

different basis sets are designed to work with
either spherical harmonics or Cartesian angular
functions

• some programs know which angular functions to
  use for a given basis set

• other programs will use one type but default, and
  you have to specify otherwise

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Contracted Gaussian Functions
Advantages of Gaussian Orbitals
1. we can treat them mathematically
2. we can introduce nodal features
3. we can introduce angular behaviour (also true
for Slater functions)
Disadvantages of Gaussian Orbitals
1. we have to use a lot of basis functions to
reproduce the behaviour of Slater orbitals
2. contraction coefficients and orbital
exponents have to be fit to either experimental
or other data tedious work without a clear
direction for improvement
In general

• most quantum chemical codes used for studying
  molecules use contracted Gaussian basis functions

• many different types of Gaussian basis sets have
  been developed for simulating molecules

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Basis Sets

• each contracted Gaussian function represents 1
  basis function

• a basis set is a set of basis functions that a
  centered on a specific atom

• basis sets usually include at least 1 basis
  function for each type of occupied orbital on the
  atom

Example
Carbon
1s22s22p2

• 1 contracted Gaussian for the 1s orbital

• 1 contracted Gaussian for the 2s orbital

• 1 contracted Gaussian for the 2p orbital

• the 2p contracted Gaussian would be multiplied by
  a p angular function to give 3 different basis
  functions

Note that 3 p functions are included even though
there are only 2 p electrons on Carbon
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Basis Sets

• each contracted Gaussian function represents 1
  atomic orbital

• a basis set is a set of basis functions centered
  on a specific atom

• basis sets usually included at least 1 basis
  function for each type of occupied orbital on the
  atom

In general

• H, He ? 1s functions

• Li - Ne ? 1s, 2s, and 2p functions (even though
  no 2p electrons on Li and Be)

• Na - Ar ? 1s, 2s, 2p, 3s, and 3p functions (even
  though no 3p electrons on Li and Be)

these are the minimum number of basis functions
that must be included for each atom
many basis sets include multiple orbitals for
each atomic state, and some higher angular
momentum orbitals (remember these are just
mathematical functions)
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Single-Zeta Basis Sets

• single-zeta (?) basis set include one contracted
  Gaussian basis function for each occupied type of
  orbital on the atom

• these are also called minimal basis sets

STO-NG

• most common single-zeta basis set

• contracts N primitive Gaussian functions

• contraction coefficients and primitive exponents
  selected to give best fit to Slater types orbitals

• STO-3G seems to provide the best balance of
  accuracy and cost

• available for nearly all elements in the periodic
  table

• not suitable for quantitative calculations

• may be OK for qualitative work

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Multiple-Zeta Basis Sets
multiple-zeta basis set include multiple
contracted Gaussian basis functions for each
occupied type of orbital on the atom

• number of contracted Gaussians used determines
  level of zeta

e.g. 2 contracted Gaussians double-zeta
3 contracted Gaussians triple-zeta
Benefits

• each contracted Gaussian function gets a
  variational coefficient in the definition of
  molecular orbitals

• more coefficients means more variational
  flexibility to get a lower energy wavefunction

• more basis functions gives more flexibility in
  describing bonding

Drawbacks

• more basis functions and molecular orbital
  coefficients increases computational effort

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Multiple-Zeta Basis Sets
multiple-zeta basis set include multiple
contracted Gaussian basis functions for each
occupied type of orbital on the atom
?
double-zeta basis set for 2p orbitals of oxygen
r-RA
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Multiple-Zeta Basis Sets
multiple-zeta basis set include multiple
contracted Gaussian basis functions for each
occupied type of orbital on the atom
cc-pVNZ

• basis functions developed to reproduce the
  results of calculations performed with
  highly-correlated methods

• use N contracted Gaussian functions per atomic
  orbital

• these basis sets are used for very accurate
  calculations, but high values of N lead to an
  impractical number of basis functions

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Split Valence Basis Sets
apply multiple-zeta basis functions to valence
states and single-zeta basis functions to core
states
rationale

• core states are relatively independent of the
  chemical environment, and dont require very much
  flexibility in their description

• valence states can participate in a wide range of
  bonding environments, and need the flexibility
  offered by multiple-zeta basis functions

• this approach strikes a balance between having a
  flexible description of the electronic structure
  and computational efficiency

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Split Valence Basis Sets
apply multiple-zeta basis functions to valence
states and single-zeta basis functions to core
states
G means Gaussian basis set
6-31G
6 describes core states
31 describes valence states
Notation

• indicates that each core state is represented
  with 1 contracted Gaussian basis function
  composed of 6 primitive Gaussian functions

6
31

• two digits means that each valence state is
  represented by two contracted Gaussian basis
  functions double-zeta

• 3 means one of the valence contracted Gaussian
  basis functions contains 3 primitive Gaussian
  functions

• 1 means the other valence contracted Gaussian
  basis function contains 1 primitive Gaussian
  function

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Split Valence Basis Sets
apply multiple-zeta basis functions to valence
states and single-zeta basis functions to core
states
3-21G

• double-zeta split valence basis set

• each core state is treated with 1 contracted
  Gaussian composed of 3 primitive Gaussians

• each valence state is treated with 2 contracted
  Gaussians

• first valence contracted Gaussian contains 2
  primitive Gaussians

• second valence contracted Gaussian contains 1
  primitive Gaussian

6-311G

• triple-zeta split valence basis set

• each core state is treated with 1 contracted
  Gaussian composed of 6 primitive Gaussians

• each valence state is treated with 3 contracted
  Gaussians

• first valence contracted Gaussian contains 3
  primitive Gaussians

• second valence contracted Gaussian contains 1
  primitive Gaussian

• third valence contracted Gaussian contains 1
  primitive Gaussian

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Polarization Functions
it is often necessary to include basis functions
with higher angular momentum than the electrons
in the atom
Examples of situations

• describing carbon in a trigonal bipyramidal state
  (e.g. transition state for the SN2 reaction from
  assignment 1) really needs orbitals of d
  symmetry

C

• p orbitals on H can account for differences in
  electron density along x and y directions when H
  is involved in bonds

H
y
x
NOTE In neither case do we suggest that d
orbitals on carbon or p orbitals on hydrogen
actually play a role in bonding. Polarization
functions just increase our mathematical
flexibility when describing the electronic
structure around a particular atom.
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Polarization Functions
it is often necessary to include basis functions
with higher angular momentum than the electrons
in the atom
Polarization functions

• H, He ? p functions

• Li Ne, Na Ar ? d functions

• transition metals ? f functions

• can also include even higher angular momenta, but
  usually less helpful

• usually designated with or (d,p)

• or (d) means polarization functions are
  added to all atoms except H and He

• or (d,p) means polarization functions are
  added to all atoms including H and He

• usually only one set of contracted Gaussians are
  added as polarization functions regardless of
  level of zeta

• cc-pVNZ basis sets include polarization functions
  by construction

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Polarization Functions
it is often necessary to include basis functions
with higher angular momentum than the electrons
in the atom
Example 6-311G for oxygen with Cartesian
Gaussians

• triple-zeta split valence basis set

• core state 1s

• 1 contracted function with 6 primitive Gaussians

• 1 contracted function with 3 primitive Gaussians
  2 contracted functions with 1 primitive
  Gaussian each

• valence state 2s

• 3 contracted functions with 3 primitive Gaussians
  each 6 contracted functions with 1 primitive
  Gaussian each

• valence state 2p

• 6 contracted functions with 1 primitive Gaussians
  each

• polarization functions d

Total 1 1 2 3 6 6 19 contracted
basis functions
Total 6 3 2 9 6 6 32 primitive
Gaussian functions
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Diffuse Functions

• sometimes electrons are not localized close to
  atoms

• so we use basis functions that decay very slowly
  with r

• these are called diffuse functions

Diffuse functions

• usually one set of diffuse functions is added for
  each occupied angular momenum in the atom

• e.g. H gets s diffuse functions, C gets s and p
  diffused functions

• these types of basis functions are needed for
  situations where electrons are not tightly bound
  to nuclei or where long-range interactions are
  relevant anions, complex, excited states

• designated with or aug

• 6-31G(d,p) includes diffuse functions on
  everything but H and H (and polarization
  functions on all atoms)

• 6-31G(d,p) includes diffuse functions and
  polarization functions on all atoms

• aug-cc-pVDZ includes diffuse functions on all
  atoms

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Diffuse Functions

• sometimes electrons are not localized close to
  atoms

• so we use basis functions that decay very slowly
  with r

• these are called diffuse functions

?
double-zeta diffuse basis set for 2p orbitals
of oxygen
diffuse function decays very slowly
r-RA
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How Many Basis Functions Am I Using?
you have to add them up
Example 6-31G(d,p) for water with Cartesian
Gaussians
Oxygen

• 1 s-type basis function with 6 primitive
  Gaussians in core

• 1 s-type basis function with 3 primitive
  Gaussians in valence

• 1 s-type basis function with 1 primitive Gaussian
  in valence

• 3 p-type basis functions with 3 primitive
  Gaussians in valence

• 3 p-type basis functions with 1 primitive
  Gaussian in valence

• 6 d-type polarization functions with 1 primitive
  Gaussian

• 1 s-type diffuse function with 1 primitive
  Gaussian

• 3 p-type diffuse functions with 1 primitive
  Gaussian

Hydrogen

• 1 s-type basis function with 3 primitive
  Gaussians in valence

• 1 s-type basis function with 1 primitive Gaussian
  in valence

• 3 p-type polarization functions with 1 primitive
  Gaussian

Total 29 basis functions and 45 primitive
Gaussians
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So, What Should I Use?
you want to use the minimum number of basis
functions to accurately describe your system
In general, the only way to find out the
appropriate basis function is to do test
calculations. The literature can also provide
guidance.
General guidelines

• 3-21G can be a good starting point for
  preliminary qualitative calculations

• polarization functions are always required for
  quantitative work

• 6-31G(d,p) is a pretty good first guess at
  choosing a basis set

• going to 6-311G(d,p) is often a waste of time

• the cc-pVNZ series of basis functions can be
  useful for highly accurate work, particularly if
  you are using configuration interaction,
  coupled-cluster or CASSCF

• diffuse functions are necessary if describing
  long-range interactions, excited states or
  anions, otherwise they are unnecessary and often
  cause numerical problems

• most basis sets were optimized to work with
  Hartree-Fock and ab initio methods, but they seem
  to work well with DFT

• most programs have a built-in library of basis
  sets, but if you dont find what you need try
  downloading it from www.emsl.pnl.gov/forms/basisf
  orm.html

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