Transcript Slide 1

Statistical Mechanics and MultiScale Simulation Methods
ChBE 591-009
Prof. C. Heath Turner
Lecture 05
• Some materials adapted from Prof. Keith E. Gubbins: http://gubbins.ncsu.edu
• Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm
Open-Shell Systems
Spin-restricted HF (RHF) = uses same spatial functions for both the a and
b spins.
Spin-unrestricted HF (UHF) = uses separate spatial functions for the a
and b spins, resulting in two Fock matrices:
Fa  ...
Fb  ...
** electron spin everywhere is NO
LONGER zero. A spin density can
now be defined:
 spin (r)   a (r)   b (r)
When is UHF needed?
• open-shell systems
• radicals
• molecules undergoing dissociation
• some ground-state molecules
Open-Shell Systems
Spin-unrestricted HF (UHF) - visualization
Advanced Ab Initio Methods
• Configuration Interaction
• Møller-Plesset perturbation theory
• Coupled-Cluster Theory
• Gaussian-2/Gaussian-3
• ONIOM
Advanced Ab Initio Methods
Electron Correlation
• HF: e- move in an average potential. Dynamics of e- neighbors (called
‘dynamical correlation’) are neglected.
• Fock operator = 1e- operator  1e- MOs  wave function (Slater
determinant)
• Ecorr = E – EHF
• Example: H2  H + H, w/o e- correlation e- are predicted to spend equal
time on both nuclei, even at infinite H-H separation.
• wavefunction is a single determinant – sometimes too rigid…
Example: Degenerate Orbitals
• Alternate approach: construct the wavefunction using multiple determinants
(ci = weight factors, subject to normalization)
  c0HF  c11  c22  
Beyond HF: Configuration Interaction
The CI method employs a wavefunction which is constructed of a
linear combination of the HF wavefunction with excited determinants
 = Co
CI
HF
+
C  +   C  + ...
a
occ vir
i
a
i
a
i
occ vir
ij
ab
ij
ab
ab
ij
The expansion coefficients are then selected so that they
variationally minimize the expectation value of the electronic energy
with respect to the CI wavefunction
CI
CI
CI
E = < |H| >
where the wavefunction is restricted by the normalization condition
1 = < | > = C +  
C ++C
a
CI
CI
2
o
occ vir
i
a2
i
occ vir
ij
ab
ab 2
ij
+ ...
Configuration Interaction for Singles
If we imagine the case of H2 we can implement CI for
singles by mixing in the excited state configurations. In
molecules of high symmetry only the configurations of
appropriate symmetry
can “mix” in this way.
CI that includes singles
only is appropriate for
improving transition
energies, but does
not help ground
state properties.
Ground
State
Excited
State
Multielectron Configuration Interaction
• The CI expansion is variational and, if the expansion is complete
(Full CI), gives the exact correlation energy (within the basis set
approximation).
• The number of determinants in Full CI grows exponentially with
the system size, making the method impractical for all but the
smallest systems. For this reason the CI expansion is usually
truncated at some order, for example CISD, where only singly
and doubly excited determinants are considered.
• Brillouin's Theorem states that singly excited determinants do not
mix with the HF determinant. Therefore CISD is the cheapest
worthwhile form of CI, yet this method scales as O(N6) where N
is the size of the system.
• CI is NOT size consistent: (EA+EB)CI ≠ (EA)CI+(EB)CI
• QCISD = size-consistent CISD theory
Perturbation Theory
Møller-Plesset perturbation theory (MPn)
** Used to account for e- correlation:
1. Create a more tractable operator.
2. Using exact eigenfunctions and eigenvalues of simplified operator,
estimate the eigenfunctions and eigenvalues of the more complete
operator.
( 0)
3. Complete operator = H (Hamiltonian): H  H  V
n
H
  fi
( 0)
i 1
4.
Correction term to the approximate Hamiltonian = difference b/t
true Hamiltonian (1st term) and Fock operator (2nd term):
1 N
V      J i  K j 
i 1 j i 1 rij
j 1
N
5.
N
The wave function and corresponding energy can be found:
( 0)
 i
( 0)
 Ei
i  i
Ei  Ei
(1)
(1)
 2 i
 2 Ei
( 2)
( 2)


Perturbation Theory
Møller-Plesset perturbation theory (MPn)
The various energy values can be calculated if the wavefunctions are known:
( 0)
i
E
   H 0  d
( 0)
i
( 0)
i
E
(0)
0

occupied

i 1
i
Ei(1)   i( 0 )V i( 0 )d
E0( 0 )  E0(1)  E0HF
Ei( 2 )   i( 0 )V i(1)d
E0( 0 )  E0(1)  E0( 2 )  E0MP 2
Ei( 3)   i( 0)V i( 2 )d
** The higher order wave functions developed by promoting electrons into
the virtual orbitals taken from the HF calculation.
MP2 = double excitations
MP3 = little improvement over MP2
MPn = size independent, not variational
Other Methods…
•
•
•
Coupled-cluster (CC) theory – a perturbation theory of
e- correlation with an excited configuration that is
“coupled” to the reference configuration.
– CCD = include double excitations
– CCSD = single + double excitations
– CCSD(T) = includes triples (approximately)
Gaussian-2/Gaussian-3 (G2/G3) = composite methods
using various levels of theory to compute
thermochemistry
ONIOM = model critical parts of the system with various
levels of theory
Density Functional Theory
•
•
•
•
•
•
Increased in popularity within last 2 decades.
Given a known e- density  form the H operator 
solve the Schrödinger Eq.  determine the
wavefunctions and energy eigenvalues.
HF – the wavefunction is essentially uninterpretable,
lack of intuition.
Hamiltonian depends ONLY on the positions and
atomic number of the nuclei and the number of e-.
HF – optimize the e- wavefunction
DFT – optimize the e- density
Definitions
Function: a prescription which maps one or more numbers
to another number: y = f(x) = x2.
Operator: a prescription which maps a function onto another
function:
2
2


F = 2  F f (x) = 2 f (x)
x
x
Functional: A functional takes a function as input and gives
a number as output. An example is:
F[f(x)] = y
Here f(x) is a function and y is a number.
An example is the functional to integrate x from –to .

F[ f ] =
f (x)dx
–