lecture feb9
DESCRIPTION
Molecular Modelling Lecture NotesTRANSCRIPT
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CHM695Feb. 9
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UHF vs RHF: H2 dissociation
UHF
RHF (H + H+)
(H + H)E R
S=0
ROHF (Restricted openshell HF): paired electrons have the constraint of same
spatial orbitals. S2 ROHF = S(S + 1) ROHF
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Electronic Correlation
iWhile solving one-electron wfns.
only mean field is taken!
electron i electron j missing!
Electronic motion is correlated: they tend to avoid each other
(more than HF)
HF energy is higher than
actual energy
Ecorr = EHF Eexact> 0
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Configuration Interaction
linear combination of n-electron wfns.
= d0 0 + d1 1 +
coefficients excited state wfn.ground state
wfn.
=Xi=0
di i
exact if the basis set is complete
1s2 1s1 2s1, 2s2 ,
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Number of configurations for a full CI:
(2K)!
n! (2K n)!very large number even for a small system!
Many determinants will not contribute much => CI on selected electrons only
and/or selected excitations only
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CIS: CI Singles (only single excitations) CID: CI Doubles (only double excitations) CISD: CI singles & doubles Valence only CI (only valence electrons) HOMO+LUMO only CI
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=Xi=0
di i
Usual way: do independent calculations to get i
following by variationally obtaining di
MCSSF: Multiconfiguration SCF ,{cj} {di} obtained by SCF
CASSCF: Complete Acitve Space SCFMCSSF is done on selected MOs only
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Mller-Plesset Perturbation Theory (MP){di} from perturbation theory
Xi
Fi
H = H0 + V
Xi,j
1
rijXj
Jj Kj
= (0) + (1) + 2 (2) + =Xr=0
r (r)
E = E(0) + E(1) + 2E(2) + =Xr=0
rE(r)
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E(0) =D (0)|H0| (0)
EE(1) =
D (0)|V | (0)
EE(2) =
D (0)|V | (1)
EE(3) =
D (0)|V | (2)
E..
MP2
MP3
HF
(1) =Xj
c(1)j (0)j
single, double excitations to
virtual orbitals of the ground state
wfn.
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Coupled Cluster Approaches:CCSD => singles & doubles CCSD(T) => CCSD+Triples
RHF
MP2CCSD
CCSD(T)Full CI
STO-3G 6-31G(d,p) 6-31G++(d,p)| | |
exact
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Computational Time:HF ~ K4
MP2 ~ K5 CCSD ~ K6
CCSD(T) ~ K7 only for ~10 atoms
a few tens of atoms
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Performance: Energetics
Errors are due to missing correlationAtomization energy of small molecules: error ~60 kcal/molHeat of reactions: error ~5-10 kcal/mol
Hartree Fock:
Energy difference between conformers: error ~1-2 kcal/mol(due to cancellation
of errors)Dispersion is not captured by HF (Note: dispersion is due to e-
correlation)MP2
Compared to HF, error decreases by 25-50%Dispersion is well captured in MP2
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Performance: StructureHartree Fock:
Usually good. Bond distances are underestimated
Typical errors: Bond distance~0.02
Bond angles~1.5 For transition state structures:
Bond distance~0.2 correlation!
MP2Usually very good Errors decrease ~50%
Distance between non-bonded atoms is better predicted than HF
(as dispersion is accounted)
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Performance: Charge Distribution
Reproduced very well in HF as well as MP2
HF dipole moments are over estimated (10-25%) HF predicts a molecule more polar than it is.
Electrostatic potential is very good.
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Density functional theory
Density: (r) =X
i
(r) (r)
.
r
number of electrons in unit volume at r
(r1) =
Zdr2 dr3 drn d!1 d!2, , d!n
(1, 2, , n) (1, 2, , n)integrate over all the coordinates except 1
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Density is function of only 3 coordinates
Many electron wave function is a function of 4n coordinates
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f(x) = x2 function of x1 12 4
number numberfunction
What does a functional mean?
function numberfunctional
f(x) = x2
Z 10
f(x)dx Z 10x2dx =
x3
3
10
=1
3functional
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Another example: (x) E
Z (x)H (x)
E[ (x)]
Our interest:
(r) EF [(r)]
density functionalinput density ??
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Does it make sense?
number of electrons+ position of nuclei+ nuclear charges
H energy
Density has the following properties:Z(r) = n
(r) has maxima at the positions of nuclei
density at the position of nuclei has information regarding nuclear charge
(r) H energy
unique
unique
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Formally this is shown by Hohenberg and Kohn (1964)
H = T + Vee + Vne
E[(r)] = T [(r)] + Vee[(r)] + Vne[(r)]
Vne , (r)unique
This implies, (r) Hunique
or, (r) F [(r)]
E
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E[(r)] = T [(r)] + Vee[(r)] + Vne[(r)]
universal functional
external potential
Functional obeys variational theorem
E[(r)] E[0(r)]exact density
variation of
density
Only valid for the exact density functional