Chemistry 6440 / 7440
Computational Chemistry
and Molecular Modeling
MidTerm Review
Types of Molecular Models
• Wish to model molecular structure,
properties and reactivity
• Range from simple qualitative descriptions
to accurate, quantitative results
• Costs range from trivial to months of
supercomputer time
• Some compromises necessary between
cost and accuracy of modeling methods
Molecular mechanics
• Ball and spring description of molecules
• Better representation of equilibrium geometries than
plastic models
• Able to compute relative strain energies
• Cheap to compute
• Lots of empirical parameters that have to be carefully
tested and calibrated
• Limited to equilibrium geometries
• Does not take electronic interactions into account
• No information on properties or reactivity
• Cannot readily handle reactions involving the making
and breaking of bonds
Semi-empirical
molecular orbital methods
• Approximate description of valence electrons
• Obtained by solving a simplified form of the
Schrödinger equation
• Many integrals approximated using empirical
expressions with various parameters
• Semi-quantitative description of electronic
distribution, molecular structure, properties and
relative energies
• Cheaper than ab initio electronic structure
methods, but not as accurate
Ab Initio Molecular Orbital Methods
• More accurate treatment of the electronic
distribution using the full Schroedinger equation
• Can be systematically improved to obtain
chemical accuracy
• Does not need to be parameterized or calibrated
with respect to experiment
• Can describe structure, properties, energetics
and reactivity
• Expensive
Potential Energy Surfaces
• The concept of potential energy surfaces is central to
computational chemistry
• The structure, energetics, properties, reactivity, spectra
and dynamics of molecules can be readily understood in
terms of potential energy surfaces
• Except in very simple cases, the potential energy surface
cannot be obtained from experiment
• The field of computational chemistry has developed a wide
array of methods for exploring potential energy surface
• The challenge for computational chemistry is to explore
potential energy surfaces with methods that are efficient
and accurate enough to describe the chemistry of interest
• Equilibrium molecular structures correspond to the positions of the minima
in the valleys on a PES
• Energetics of reactions can be calculated from the energies or altitudes of
the minima for reactants and products
• A reaction path connects reactants and products through a mountain pass
• A transition structure is the highest point on the lowest energy path
• Reaction rates can be obtained from the height and profile of the potential
energy surface around the transition structure
• The shape of the valley around a minimum determines the vibrational
spectrum
• Each electronic state of a molecule has a separate potential energy
surface, and the separation between these surfaces yields the electronic
spectrum
• Properties of molecules such as dipole moment, polarizability, NMR
shielding, etc. depend on the response of the energy to applied electric
and magnetic fields
Asking the Right Questions
• molecular modeling can answer some
questions easier than others
• stability and reactivity are not precise
concepts
– need to give a specific reaction
• similar difficulties with other general
concepts:
– resonance
– nucleophilicity
– leaving group ability
– VSEPR
– etc.
Asking the Right Questions
• phrase questions in terms of energy differences,
energy derivatives, geometries, electron distributions
• trends easier than absolute numbers
• gas phase much easier than solution
• structure and electron distribution easier than
energetics
• vibrational spectra and NMR easier than electronic
spectra
• bond energies, IP, EA, activation energies are hard
(PA not quite as hard)
• excited states much harder than ground states
• solvation by polarizable continuum models (very hard
by dynamics)
Molecular Mechanics
• PES calculated using empirical potentials
fitted to experimental and calculated data
• composed of stretch, bend, torsion and
non-bonded components
E = Estr + Ebend + Etorsion + Enon-bond
• e.g. the stretch component has a term for
each bond in the molecule
Bond Stretch Term
• many force fields use just a quadratic term, but the
energy is too large for very elongated bonds
Estr =  ki (r – r0)2
• Morse potential is more accurate, but is usually not used
because of expense
Estr =  De [1-exp(-(r – r0)]2
• a cubic polynomial has wrong asymptotic form, but a
quartic polynomial is a good fit for bond length of interest
Estr =  { ki (r – r0)2 + k’i (r – r0)3 + k”i (r – r0)4 }
• The reference bond length, r0, not the same as the
equilibrium bond length, because of non-bonded
contributions
Angle Bend Term
• usually a quadratic polynomial is sufficient
Ebend =  ki ( – 0)2
• for very strained systems (e.g. cyclopropane) a
higher polynomial is better
Ebend =  ki ( – 0)2 + k’i ( – 0)3
+ k”i ( – 0)4 + . . .
• alternatively, special atom types may be used for
very strained atoms
Torsional Term
• most force fields use a single cosine with
appropriate barrier multiplicity, n
Etors =  Vi cos[n( – 0)]
• some use a sum of cosines for 1-fold (dipole), 2fold (conjugation) and 3-fold (steric)
contributions
Etors =  { Vi cos[( – 0)] + V’i cos[2( – 0)]
+ V”i cos[3( – 0)] }
Non-Bonded Terms
• Lennard-Jones potential
– EvdW =  4 ij ( (ij / rij)12 - (ij / rij)6 )
– easy to compute, but r -12 rises too rapidly
• Buckingham potential
– EvdW =  A exp(-B rij) - C rij-6
– QM suggests exponential repulsion better, but is harder
to compute
• tabulate  and  for each atom
– obtain mixed terms as arithmetic and geometric means
– AB = (AA + BB)/2; AB = (AA BB)1/2
Parameterization
• difficult, computationally intensive, inexact
• fit to structures (and properties) for a training set of
molecules
• recent generation of force fields fit to ab initio data
at minima and distorted geometries
• trial and error fit, or least squares fit (need to avoid
local minima, excessive bias toward some
parameters at the expense of others)
• different parameter sets and functional forms can
give similar structures and energies but different
decomposition into components
• don't mix and match
Applications
• good geometries and relative energies of conformers of the
same molecule (provided that electronic interactions are not
important)
• effect of substituents on geometry and strain energy
• well parameterized for organics, less so for inorganics
• specialty force fields available for proteins, DNA, for liquid
simulation
• molecular mechanics cannot be used for reactions that
break bonds
• useful for simple organic problems: ring strain in
cycloalkanes, conformational analysis, Bredt's rule, etc.
• high end biochemistry problems: docking of substrates into
active sites, refining x-ray structures, determining structures
from NMR data, free energy simulations
Schrödinger Equation
Ĥ  E
• H is the quantum mechanical Hamiltonian for the
system (an operator containing derivatives)
• E is the energy of the system
•  is the wavefunction (contains everything we
are allowed to know about the system)
• ||2 is the probability distribution of the particles
Hamiltonian for a Molecule
ˆ 
H
electrons

i
  2 2 nuclei   2 2 electronsnuclei  e 2 Z A electrons e 2 nuclei e 2 Z A Z B
i  
A   
 

2me
riA
rij
rAB
A 2m A
i
A
i j
A B
• kinetic energy of the electrons
• kinetic energy of the nuclei
• electrostatic interaction between the electrons
and the nuclei
• electrostatic interaction between the electrons
• electrostatic interaction between the nuclei
Variational Theorem
• the expectation value of the Hamiltonian is the
variational energy
* ˆ

 Hd
  d
*
 Evar  Eexact
• the variational energy is an upper bound to the lowest
energy of the system
• any approximate wavefunction will yield an energy
higher than the ground state energy
• parameters in an approximate wavefunction can be
varied to minimize the Evar
• this yields a better estimate of the ground state energy
and a better approximation to the wavefunction
Born-Oppenheimer Approximation
• the nuclei are much heavier than the electrons
and move more slowly than the electrons
• in the Born-Oppenheimer approximation, we
freeze the nuclear positions, Rnuc, and calculate
the electronic wavefunction, el(rel;Rnuc) and
energy E(Rnuc)
• E(Rnuc) is the potential energy surface of the
molecule (i.e. the energy as a function of the
geometry)
• on this potential energy surface, we can treat the
motion of the nuclei classically or quantum
mechanically
Hartree Approximation
• assume that a many electron wavefunction
can be written as a product of one electron
functions
(r1 , r2 , r3 ,)   (r1 ) (r2 ) (r3 )
• if we use the variational energy, solving the
many electron Schrödinger equation is
reduced to solving a series of one electron
Schrödinger equations
• each electron interacts with the average
distribution of the other electrons
Hartree-Fock Approximation
• the Pauli principle requires that a wavefunction for
electrons must change sign when any two electrons
are permuted
• the Hartree-product wavefunction must be
antisymmetrized
• can be done by writing the wavefunction as a
determinant

1 (1) 1 (2)  1 (n)
1 2 (1) 2 (2)  2 (n)
n




n (1) n (1)  n (n)
 1 2  n
Fock Equation
• take the Hartree-Fock wavefunction
  1 2  n
• put it into the variational energy expression
Evar 
*

 Ĥd
*

  d
• minimize the energy with respect to changes in the orbitals
Evar / i  0
• yields the Fock equation
F̂i   ii
Fock Operator
ˆ V
ˆ  Jˆ  K
ˆ
Fˆ  T
NE
• Coulomb operator (electron-electron repulsion)
2
e
Jˆ i  {    j  j d }i
rij
j
• exchange operator (purely quantum mechanical
-arises from the fact that the wavefunction must
switch sign when you exchange to electrons)
electrons
e2
  j rij i d } j
electrons
ˆ  {
K
i

j
Solving the Fock Equations
F̂i   ii
1. obtain an initial guess for all the orbitals i
2. use the current I to construct a new Fock
operator
3. solve the Fock equations for a new set of I
4. if the new I are different from the old I, go
back to step 2.
LCAO Approximation
•
•
•
•
numerical solutions for the Hartree-Fock
orbitals only practical for atoms and diatomics
diatomic orbitals resemble linear combinations
of atomic orbitals
e.g. sigma bond in H2
  1sA + 1sB
for polyatomics, approximate the molecular
orbital by a linear combination of atomic
orbitals (LCAO)
   c  

Roothaan-Hall Equations
•
•
•
•
•
basis set expansion leads to a matrix form of
the Fock equations
F Ci = i S Ci
F – Fock matrix
Ci – column vector of the molecular orbital
coefficients
I – orbital energy
S – overlap matrix
Slater-type Basis Functions



1/ 2

3
1s (r )   1s /  exp(  1s r )
1/ 2

5
 2 s (r )   2 s / 96 r exp(  2 s r / 2)
1/ 2

5
 2 px (r )   2 p / 32 x exp(  2 p r / 2)
•
•
•
•
•



exact for hydrogen atom
used for atomic calculations
right asymptotic form
correct nuclear cusp condition
3 and 4 center two electron integrals cannot be
done analytically
Gaussian-type Basis Functions





1/ 4

3
g s (r )  2 /  exp(  r 2 )

5
3 1/ 4
g x (r )  128 / 
x exp(  r 2 )

7
3 1/ 4 2
g xx (r )  2048 / 9
x exp(  r 2 )

7
3 1/ 4
g xy (r )  2048 / 
xy exp(  r 2 )
•
•
•



die off too quickly for large r
no cusp at nucleus
all two electron integrals can be done
analytically
Minimal Basis Set
•
•
•
only those shells of orbitals needed for a
neutral atom
e.g. 1s, 2s, 2px, 2py, 2pz for carbon
STO-3G
– 3 gaussians fitted to a Slater-type orbital (STO)
– STO exponents obtained from atomic
calculations, adjusted for a representative set of
molecules
•
also known as single zeta basis set (zeta, ,
is the exponent used in Slater-type orbitals)
Double Zeta Basis Set (DZ)
•
•
•
•
•
each function in a minimal basis set is doubled
one set is tighter (closer to the nucleus, larger
exponents), the other set is looser (further from
the nucleus, smaller exponents)
allows for radial (in/out) flexibility in describing
the electron cloud
if the atom is slightly positive, the density will
be somewhat contracted
if the atom is slightly negative, the density will
be somewhat expanded
Split Valence Basis Set
•
•
•
•
only the valence part of the basis set is
doubled (fewer basis functions means less
work and faster calculations
core orbitals are represented by a minimal
basis, since they are nearly the same in atoms
an molecules
3-21G (3 gaussians for 1s, 2 gaussians for the
inner 2s,2p, 1 gaussian for the outer 2s,2p)
6-31G (6 gaussians for 1s, 3 gaussians for the
inner 2s,2p, 1 gaussian for the outer 2s,2p)
Polarization Functions
•
•
•
•
•
•
•
higher angular momentum functions added to a basis
set to allow for angular flexibility
e.g. p functions on hydrogen, d functions on carbon
large basis Hartree Fock calculations without
polarization functions predict NH3 to be flat
without polarization functions the strain energy of
cyclopropane is too large
6-31G(d) (also known as 6-31G*) – d functions on
heavy atoms
6-31G(d,p) (also known as 6-31G**) – p functions on
hydrogen as well as d functions on heavy atoms
DZP – DZ with polarization functions
Diffuse Functions
•
•
•
•
functions with very small exponents added to a
basis set
needed for anions, very electronegative atoms,
calculating electron affinities and gas phase
acidities
6-31+G – one set of diffuse s and p functions
on heavy atoms
6-31++G – a diffuse s function on hydrogen as
well as one set of diffuse s and p functions on
heavy atoms
Correlation-Consistent
Basis Functions
•
•
•
•
•
•
a family of basis sets of increasing size
can be used to extrapolate to the basis set limit
cc-pVDZ – DZ with d’s on heavy atoms, p’s on H
cc-pVTZ – triple split valence, with 2 sets of d’s
and one set of f’s on heavy atoms, 2 sets of p’s
and 1 set of d’s on hydrogen
cc-pVQZ, cc-pV5Z, cc-pV6Z
can also be augmented with diffuse functions
(aug cc-pVXZ)
Molecular Orbital Plots


i (r )   ci   (r )

• plot a surface where
|i(r)|2 = c
• i(r) can have positive
and negative values
• shade in different
colors
• only the change in
sign matters, not the
absolute sign
Population Analysis
occ
*
2
c
 i ci  P
density matrix, S  overlap matrix
i
 partition
P S   N


e
into contributi ons
from different atoms and basis functions
M   P S  Mulliken population analysis matrix
M AB   P S 
condensed to atoms
 A B
q A  Z A   M AB
B
atomic cha rge
Dipole Moment
• for Hartree-Fock wavefunctions, the dipole is the
expectation value of the classical expression for
the dipole
• can be written in terms of the density matrix and
a set of dipole integrals over the basis functions



*
   (  eri ) d   eZ A RA
i
A


  2  (er )i d   eZ A RA
occ
*
i
i
A


  P    (er )  d   eZ A RA

A
Electron Density



 (r )   P   (r )  (r )

Electrostatic Potential
• energy of a unit test charge placed at rC



*
ESP(rC )   (e / rC ) d   eZ A / RAC
A


  P    (e / rC )  d   eZ A / RAC

A
Features of Potential Energy
Surfaces
Initial guess for geometry & Hessian
Calculate energy and gradient
Minimize along line between
current and previous point
Update Hessian
(Powell, DFP, MS, BFGS, Berny, etc.)
Take a step using the Hessian
(Newton, RFO, Eigenvector following)
Check for convergence
on the gradient and displacement
no
Update the geometry
yes
DONE
Testing Minima
• Compute the full Hessian (the partial Hessian
from an optimization is not accurate enough and
contains no information about lower symmetries).
• Check the number of negative eigenvalues:
– 0 required for a minimum.
– 1 (and only 1) for a transition state
• For a minimum, if there are any negative
eigenvalues, follow the associated eigenvector to
a lower energy structure.
• For a transition state, if there are no negative
eigenvalues, follow the the lowest eigenvector up
hill.
Algorithms for Finding
Transition States
•
•
•
•
Surface fitting
Linear and quadratic synchronous transit
Coordinate driving
Hill climbing, walking up valleys, eigenvector
following
• Gradient norm method
• Quasi-Newton methods
• Newton methods
Gradient Based Transition Structure
Optimization Algorithms
• Quadratic Model
– fixed transition vector
– constrained transition vector
– associated surface
– fully variable transition vector
• Non Quadratic Models-GDIIS
• Eigenvector following/RFO for stepsize control
• Bofill update of Hessian, rather than BFGS
• Test Hessian for correct number of negative
eigenvalues
Testing Transition Structures
• Compute the full Hessian (the partial Hessian from an
optimization is not accurate enough and contains no
information about lower symmetries).
• Check the number of negative eigenvalues:
– 1 and only 1 for a transition state.
• Check the nature of the transition vector (it may be
necessary to follow reaction path to be sure that the
transition state connects the correct reactants and
products).
• If there are too many negative eigenvalues, follow the
appropriate eigenvector to a lower energy structure.
Reaction Paths
Taylor expansion of reaction path
0
2 1
3 2
x(s)  x(0)  s (0)  1 2 s  (0)  1 6 s  (0)  
d x( s )  g
 

ds
|g|
0
Tangent
d 0 ( s ) d 2 x( s )
1 

ds
d s2
Curvature
1
  (H 0  ( 0tH 0 ) 0 ) / | g |
Harmonic Vibrational Frequencies
for a Polyatomic Molecule
ˆ
H
nuc
 2 2 1 2

 qi
2
2 q i 2
i, j
i
~
t
  L k L  L M k ML  i 
2
q  Lt  Lt Mx M i , j   i , j / mi
t
I – eigenvalues of the mass weighted Cartesian
force constant matrix
qi – normal modes of vibration
Calculating Vibrational Frequencies
• optimize the geometry of the molecule
• calculate the second derivatives of the HartreeFock energy with respect to the x, y and z
coordinates of each nucleus
• mass-weight the second derivative matrix and
diagonalize
• 3 modes with zero frequency correspond to
translation
• 3 modes with zero frequency correspond to overall
rotation (if the forces are not zero, the normal
modes for rotation may have non-zero frequencies;
hence it may be necessary to project out the
rotational components)
Pople, J. A.; Schlegel, H. B.; Krishnan, R.; DeFrees, D. J.; Binkley, J. S.; Frisch, M. J.;
Whiteside, R. A.; Hout, R. F.; Hehre, W. J.; Molecular orbital studies of vibrational
frequencies. Int. J. Quantum. Chem., Quantum Chem. Symp., 1981, 15, 269-278.
Scaling of Vibrational Frequencies
• calculated harmonic frequencies are typically 10%
higher than experimentally observed vibrational
frequencies
• due to the harmonic approximation, and due to the
Hartree-Fock approximation
• recommended scale factors for frequencies
HF/3-21G 0.9085, HF/6-31G(d) 0.8929,
MP2/6-31G(d) 0.9434, B3LYP/6-31G(d) 0.9613
• recommended scale factors for zero point energies
HF/3-21G 0.9409, HF/6-31G(d) 0.9135,
MP2/6-31G(d) 0.9676, B3LYP/6-31G(d) 0.9804
Electron Correlation Energy
• in the Hartree-Fock approximation, each electron
sees the average density of all of the other
electrons
• two electrons cannot be in the same place at the
same time
• electrons must move two avoid each other,
i.e. their motion must be correlated
• for a given basis set, the difference between the
exact energy and the Hartree-Fock energy is the
correlation energy
• ca 20 kcal/mol correlation energy per electron pair
Goals for Correlated Methods
• well defined
– applicable to all molecules with no ad-hoc choices
– can be used to construct model chemistries
• efficient
– not restricted to very small systems
• variational
– upper limit to the exact energy
• size extensive
– E(A+B) = E(A) + E(B)
– needed for proper description of thermochemistry
• hierarchy of cost vs. accuracy
– so that calculations can be systematically improved
Configuration Interaction
• determine CI coefficients using the variational principle
  0   tia ia   tijab ijab 
ia
ijab
abc
abc
t

 ijk ijk  
ijkabc
ˆ d /  *d with respect to t
minimize E    *H

• CIS – include all single excitations
– useful for excited states, but on for correlation of the ground state
• CISD – include all single and double excitations
– most useful for correlating the ground state
– O2V2 determinants (O=number of occ. orb., V=number of unocc. orb.)
• CISDT – singles, doubles and triples
– limited to small molecules, ca O3V3 determinants
• Full CI – all possible excitations
– ((O+V)!/O!V!)2 determinants
– exact for a given basis set
– limited to ca. 14 electrons in 14 orbitals
Møller-Plesset Perturbation Theory
• choose H0 such that its eigenfunctions are
determinants of molecular orbitals
ˆ   Fˆ
H
0
i
• expand perturbed wavefunctions in terms of
the Hartree-Fock determinant and singly,
doubly and higher excited determinants
1   aia ia   aijab ijab 
ia
ijab
abc
abc
a

 ijk ijk  
ijkabc
• perturbational corrections to the energy
ˆ  d   V
ˆ  d
EHF  E0  E1   0 H
0 0
0
0

ˆ  d  E 
EMP 2  EHF  E2  EHF   0 V
1
HF

i  j , a b
ˆ  ab d ]2
[  0 V
ij
a  a  i   j
Coupled Cluster Theory
• CISD can be written as
ˆ T
ˆ )
CISD  (1  T
1
2
0
• T1 and T2 generate all possible single and
double excitations with the appropriate
coefficients
T̂2 0 
ab
ab
t

 ij ij
i  j , a b
• coupled cluster theory wavefunction
ˆ T
ˆ )
CCSD  exp(1  T
1
2
0
Theoretical Basis for
Density Functional Theory
• Hohenberg and Kohn (1964)
–
–
–
–
energy is a functional of the density E[]
the functional is universal, independent of the system
the exact density minimizes E[]
applies only to the ground state
• Kohn and Sham (1965)
– variational equations for a local functional
E[  ]  T [  ]  VNE [  ]  J [  ]  Exc [  ]  Vnuc
– density can be written as a single determinant of orbitals (but
orbitals are not the same as Hartree-Fock)
– EXC takes care of electron correlation as well as exchange
Density Functional Theory
• local functionals (LSDA)
– depend only on the density
– exchange and correlation functional from electron gas
• generalized gradient approximation (GGA)
– depends on ||/4/3
– BLYP, BP86, BPW91, PBE
• hybrid functionals
– mix some Hartree-Fock exchange
– B3LYP, PBE1PBE, B3PW91
Semi-empirical MO Methods
• the high cost of ab initio MO calculations is largely
due to the many integrals that need to be
calculated (esp. two electron integrals)
• semi-empirical MO methods start with the general
form of ab initio Hartree-Fock calculations, but
make numerous approximations for the various
integrals
• many of the integrals are approximated by
functions with empirical parameters
• these parameters are adjusted to improve the
agreement with experiment
Zero Differential Overlap (ZDO)
•
two electron repulsion integrals are one of the
most expensive parts of ab initio MO
calculations
(  |  )     (1)  (1)
•
•
•
1
  (2)  (2)d 1d 2
r12
neglect integrals if orbitals are not the same
(  |  )  (  |  )    
where    1 if    ,    0 if   
approximate integrals by using s orbitals only
CNDO, INDO and MINDO semi-empirical
methods
Neglect of Diatomic Differential
Overlap (NDDO)
•
fewer integrals neglected
1
(  |  )     (1)  (1)   (2)  (2)d 1d 2
r12
•
neglect integrals if  and  are not on the same atom or
 and  are not on the same atom
integrals approximations are more accurate and have
more adjustable parameters than in ZDO methods
parameters are adjusted to fit experimental data and ab
initio calculations
MNDO, AM1 and PM3 semi-empirical methods
•
•
•
Model Chemistries
• A theoretical model chemistry is a complete algorithm for
the calculation of the energy of any molecular system.
• It cannot involve subjective decisions in its application.
• It must be size consistent so that the energy of every
molecular species is uniquely defined.
• A simple model chemistry employs a single theoretical
method and basis set.
• A compound model chemistry combines several
theoretical methods and basis sets to achieve higher
accuracy at lower cost.
• A model chemistry is useful if for some class of molecules
it is the most accurate calculation we can afford to do.
Model Chart
Minimal
STO-3G
HF
MP2
MP3
MP4
QCISD(T)
...
Full CI
Split-Valence
3-21G
Basis
Polarized
6-31G*
6-311G*
Diffuse
6-311+G*
High Ang. Mom.
6-311+G(3df,p)
…

Schrödinger
Equation
Development of a
Model Chemistry
• Set targets
– accuracy goals
– cost/size goals
– validation data set
• Define and implement methods
– Specify level of theory for geometry optimization,
electronic energy, vibrational zero point energy
• Test model on validation data set
Compound Model
Chemistries:
G2 and G2(MP2)
• Proposed by J. Pople and co-workers
• Goal: Atomization energies to 2 kcal/mol
• Strategy: Approximate QCISD(T)/6311+G(3df,p) by assuming that basis set
and correlation corrections are additive
• Mean absolute error of 1.21 kcal/mol in
125 comparisons
CBS Extrapolation
The slow, N-1, convergence
of the correlation energy vs
the one-electron basis set
expansion is the result of
the universal cusp in wave
functions as interelectronic
distances, rij  0 .
Thus, we can reasonably
expect the N-1 form to also
be universal.
SCF
MP2
MP4(SDQ) MP4(SDTQ) QCISD(T)
FCI
6-31G
631G†
6-31+G†
6-31+G††
6-311G(d,p)
6-31+G(d(f),d,p)
6-311+G(d,p)
6-311G(2df,p)
6-311+G(2df,p)
6-311+G(3df,2p)
6-311+G(3d2f,2df,p)
6-311++G(3d2f,2df,2p)
[6s6p3d2f,4s2p1d]
CBS
Exact
RM S Error: G 2 test set (kcal/m ol)
3.0
2.5
C BS-4
C BS-q
2.0
G 2(MP2)
G2
1.5
C BS-Q B3
1.0
C BS-Q C I/APNO
0.5
0.0
0
5
10
15
20
25
M axim um N um ber of H eavy Atom s
30
Thermodynamic Functions
• U(T) - internal energy at absolute
temperature T
• H(T) = U(T) + PV = U(T) + RT - enthalpy
• S(T) - entropy
• G(T) = H(T) – T S(T) – free energy
Thermodynamic Functions
• at absolute zero, T = 0
U(0) = H(0) = G(0)
U(0) = electronic energy
+ zero point energy
S(0) = 0 for a pure crystalline substance
(third law of thermodynamics)
Thermodynamic Functions at T  0
• U(T) = U(0) + CvdT
– heat at constant volume, molecule gains energy
for translation (3/2 RT), rotation (3/2 RT) and
vibration ( 1/(1-exp(-i/kT))
• H(T) = H(0) + CpdT
– heat at constant pressure, molecule gains
additional energy from expansion
• S(T) > 0
– more states become accessible as the
temperature increases