Semi-­‐Empirical Methods CHEM 430 Spring 2016 Cost Correlated Method (MP2, CCSD) AtomisFc MM Molecular detail •  Hartree-­‐Fock scales as N4 (N=# basis funcFons) –  Due to two-­‐
electron integrals within Fock matrix •  Semi-­‐empirical cut cost by reducing number of integrals –  Typically scale as N2 100 fs, 10
atoms:
photochemistry
10 ms, thousands
of atoms:
protein folding,
drug binding
Density FuncFonal Theory 10 ps, 100
atoms: chemical
reactions
Coarse-­‐grain 1 ms+, 1 million
atoms: dynamics of
large proteins, cell
membranes, viruses
ComputaFonal cost 2 How to cut cost •  Consider only valence electrons –  Account for core by reducing nuclear charge or introducing funcFons to model the combined repulsion due to the nuclei and core electrons •  Only use a minimum basis set for the valence electrons –  Hydrogen has one basis funcFon –  Second and third row atoms has four basis funcFons (s, px, py, pz) •  Neglect integrals or fit empirical funcFons to approximate the integrals –  Adjust parameters to fit experiment 3 Zero DifferenFal Overlap (ZDO) •  Neglect all products of basis funcFons depending on the same electron coordinates when located on different atoms (µv | λσ ) =
∫
χ µ (1)χ v (1)
1
χ λ (2) χ σ (2)dτ 1dτ 2
r12
(µv | λσ ) = (µµ | λλ ) δµvδλσ
Neglect integrals if orbitals not the same where δµv = 1 if µ = v, δµv = 0 if µ ≠ v
•  Consequences: –  Overlap matrix S is a unit matrix –  One-­‐electron integrals involving three centers are set to zero –  All three-­‐ and four-­‐center two-­‐electron integrals (by far the most numerous of the two-­‐electron integrals) are neglected •  Remaining integrals are made into parameters and fit to experimental and calculated data –  Semi-­‐empirical methods defined by what integrals are neglected and what data the parameters are fit to 4 Neglect of Diatomic DifferenFal Overlap (NDDO) •
•
•
•
No further approximaFons than ZDO Integral approximaFons are more exact than ZDO More adjustable parameters than ZDO Introduced by John Pople 5 Intermediate Neglect of DifferenFal Overlap (INDO) •  Neglects all two-­‐center two-­‐electron integrals which are not of the Coulomb type •  Preserves rotaFonal invariance –  Total energy independent of rotaFon of coordinate system àSome integrals made independent of orbital type (e.g., integral involving a p-­‐orbital must be the same as with an s-­‐orbital) àOne-­‐electron integrals with two different funcFons on the same atom and a potenFal energy operator from another atom disappear •  Happy medium between NDDO and CNDO 6 Complete Neglect of DifferenFal Overlap (CNDO) •  Only Coulomb one-­‐center and two-­‐center two-­‐electron integrals remain •  Integrals and approximaFons the same as in INDO CNDO vs INDO vs NDDO Only differ in treatment of two-­‐electron integrals CNDO, INDO Reduce two-­‐electron integrals to two parameters, γAA and γAB NDDO All one-­‐ and two-­‐center integrals are kept 7 Modified NDDO model •  Modified neglect of differenFal overlap (MNDO) •  AusFn Model 1 (AM1) •  Parametric Method Number 3 (PM3) How to transform NDDO into working computaFonal model… •  Remaining integrals can be calculated from funcFonal form of the atomic orbitals •  Remaining integrals can be made into parameters (and then fit to experimental data) •  Remaining integrals can be made into parameters (assigned values based on experimental data) 8 MNDO •  InteracFons between O-­‐H and N-­‐H bonds treated differently " e −α R
%
MNDO
'
−α R
Vnn (A, H ) = Z A Z H sA sH sA sH $1+
+e
'
R
#
&
AH
•  Core-­‐core repulsion treated differently A AH
H
AH
Fit the α values VnnMNDO (A, B) = Z A' Z B' sA sB sA sB (1+ e−α A RAB + e−α B RAB )
•  Fit for elements H,B,C,N,O,F,Al,Si,P,S,Cl,Zn,Ge,Br,Sn,I,Hg,Pb •  Parameters taken from atomic spectra and others fit to molecular data Limita0ons •  Sterically crowded molecules, four membered rings, weak interacFons, acFvaFon energies, ethers, sulfides 9 AM1 •  MNDO repulsion between two atoms which are 2-­‐3 Angstroms apart is too highàacFvaFon energies too large (among other errors)…due to too repulsive interacFon in the core-­‐core potenFal •  Modified core-­‐core funcFon from MNDO by adding Gaussian funcFons and reparameterized AM1
nn
V
MINDO
nn
(A, B) = V
&
Z A' Z B' #
−bkA ( RAB −ckA )2
−bkB ( RAB −ckB )2
(A, B) +
α
e
+
α
e
%∑ kA
(
∑ kB
RAB $ k
'
k
•  k between 2 and 4 depending on the atom •  ak,bk,ck fit to molecular data LimitaFons •  Geometries with hydrogen bonds, alkyl groups, nitro compounds, peroxide bonds, phosphor compounds 10 PM3 •  Same core-­‐core repulsion as AM1 but only two Gaussian funcFons added to each atom •  Automated opFmizaFon process –  All parameters opFmized simultaneously (MNDO, AM1 done by hand) –  Significantly larger training set used LimitaFons •  Hydrogen bonds too short, gauche of ethanol more stable than trans, Si-­‐X bonds too short, charge of nitrogen atoms ohen incorrect sign and/or magnitude 11 ParameterizaFon Performance of semi-­‐empirical methods 13 More performance discussion 14 More performance discussion 15