A BZIS FGGVNY HIBA S KORRELCIS ENERGIA GYORS BECSLSE PARCILIS TLTSEKBOL REBECEP MDSZER BEMUTATSA Kri

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May 21-26, 2006ICQC-XII, KYOTO, JAPAN
STATISTICAL ANALYSIS OF THE DEPENDENCE
OFCORRELATION ENERGY AND ZERO POINT ENERGYON
THE NUCLEAR FRAME AND NUMBER OFELECTRONS IN
MOLECULAR SYSTEMS Sandor Kristyan Hungarian
Academy of Sciences, Central Institute of
Chemistry, 1025. Budapest, Pusztaszeri ut
59-67, HUNGARY
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Schrödinger equation HYEY Born-Oppenheimer
approx. H H(el) H(nucl),
(m(proton)/m(electron) ? 1840). 1.
Mapping of electronic potential surfaces (non
rel., spin-less, fixed nuclear config.).
Pauli anti-symmetry principle on Yel
2. Reaction kinetics (ab initio, RRKM, etc.)
Introduction
H3 pot. Surf. (hyper-spherical coord.)
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Mapping of electronic potential
surfaceMolecular mechanics many
empirical parameters, no bond breaking/formation,
fast, low disc space demandAb initio method
(numerical, no empirical parameter (only
h(Planc), m(electr.), etc.) HF-SCF
ground state, closed (and open) shell, around
stationary point, short calculation,
low disc space demand Yel one Slater
determinant, variational principle CI
ground- and excited states, closed (and
open) shell, any nucl. config. (stac. point and
van der Waals region, as well as
far distance nucl. dist. (size consistency),
long calc., large disc space demand
full-CI, MRDCI, etc. Yel many Slater
determinant ...
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Error of HF-SCF is called the correlation
energy
E0(CI) E0(HF-SCF/basis) Ecorr
containing the basis set error
methods MP2, coupled cluster (CCSD,), G2-G3,
DCI, CAS, etc.
As well as DFT (Kohn Sham equations)
Time and disc space demand of these correlation
calculations are 2-1000 -times more than
the needs of HF-SCF procedure, as well as
more complex programming and software use.
(Except e.g. B3LYP)
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Example Methane (CH4), Gaussian98-al
(Mulliken analysis), Silicon Graphics (175 MHz)
machine Methane (CH4), Td simetry
MP2(full)/6-31G Geom. E0(HF-SCF/STO-3G)
-39.72672 hartree, 4.7sec/10MB/27 pr.g.
E0(HF-SCF/6-31G) -40.19507 hartree,
5.7sec/10MB/44 pr.g. E0(MP2)
-40.33704 hartree E0(G2)
-40.45354 hartree, 22.9sec/40MB/102 pr.g.
Ecorr(STO-3G,G2) -40.45354(-39.72672)
-0.72682 hartree Ecorr(6-31G,G2)
-40.45354(-40.19507) -0.25847 hartree
(1 hartree 627.5 kcal/mol, pr.g.
primitive gaussian)
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For energy differences the big part of the error
of HF-SCF drops, but the chemical accuracy (i.e.
the 1 kcal/mol) is not guaranted ? accurate
correlation calculation is necessary and
accurate ZPE calculation as well. In
chemistry the range of potential barriers are
0-5 kcal/mol (compare chemical accuracy and
correlation energy).
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Statistics 1 Ecorr vs. a(N-1) and the method
(REBECEP, REZEP)
A simple, quasi-linear expression between the
correlation energy (Ecorr) and the number of
electrons (N) in a molecular system 1-3 (for
energy diff. this correction drops
!) -0.030(N-1) gt Ecorrhartree gt -0.045(N-1)
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The development of this a(N-1) relation
1-3, 7 has been done by weighting with partial
charges on atoms in the molecule
Partial charges from HF-SCF calculation.
Chemical accuracy has been approached. THIS
METHOD IS A DFT METHOD, BUT BASICALLY DIFERS FROM
THE CONVENTIONAL (E.G. KOHN-SHAM, EXCHNG-CORR.
FUNCTIONAL) FORMALISM.
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Popular partial charge definitions (the
definition is never unique) ChelpG,
Merz-Kollman (MK), Mulliken (overlap
integrals, simple, problem of large basis set),
Natural population analysis (NPA, long ),
Stockholder (with respect to atoms ),
Electrostatic charges (with classic point charge
), Löwdin bond order (partial charge can be
defined ), Bader charges (along the
gradient of electron density , difficult,
sometimes divergent, regions
without nucleus can occur), etc.
Methyl pyruvate (piroszolosav), electron density
equipotential surface (0.002)
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REBECEP APPROACHES THE CHEMICAL
ACCURACY! Restrictions Only for stationary
points (as the HF-SCF), now worked out for
minimums. Open and closed shells must be
treated differently, now worked out for closed
shells. Currently we have parameters for H,
C, N, O and F atoms, but it can be expanded. CPU
time and disc space demand what the
E0(HF-SCF/basis/parc.charge) has, the
calculation of Ecorr(RECEP/basis/parc.charge/G2
quality) is instant. It can be done even
only on a pocket calculator !!! The
Ecorr(RECEP) equations are liner gt multi-linear
parameter fit is possible. Other properties
Hypothesis parameters can be obtained for any
partial charge. Mulliken charge is the most
practical, because simple and available (e.g.
Gaussian94,98, etc.).
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REBECEP ABACUS 1-3, 7 THIS TABLE SHOWS THE
ENTIRE REBECEP PROCEDURE!
Calculation of Ecorr for methyl-nitrit
(CH3-O-NO), fitted REBECEP atomic correlation
parameters (G2 correlation energies and
HF-SCF/6-311G(2d,p) NPA partial charges) ZA
Partial.chrg.A N1 NA N2 Ecorr(N1,ZA)
Ecorr(N2,ZA) Einterpolated,A? Ecorr(NA,ZA)
6 -0.133 6 6.133 7 -0.1659 -0.1909 -0.1692
8 -0.490 8 8.490 9 -0.2703 -0.2790 -0.
2746 1 0.171 0 0.829 2 0.0 -0.0376 -0.
0156 1 0.165 0 0.835 2 0.0 -0.0376 -0.
0157 1 0.165 0 0.835 2 0.0 -0.0376 -0.
0157 7 0.504 6 6.496 7 -0.2227 -0.2259
-0.2243 8 -0.382 8
8.382 9 -0.2703 -0.2790 -0.2736
Ecorr(REBECEP-fit) S(A)Einterpolated,A
-0.9886 hartree (accurate up to 1 kcal/mol w/r
G2!). (It can be done even on a pocked
calculator two important modules are necessary
HF-SCF/basis and partial charge
calculation.) Further important model
development 3 Mulliken charge (simple, fast,
convergent), and the small basis (6-31G) is
enough for the accuracy. REBECEP Rapid
Estimation of Basis Set Error and Correlation
Energy from Partial Charges
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The same REBECEP formalism is suitable for
the calculation of ZPE 4. REZEP Rapid
estimation of zero point energy. (ZPE (1/2)?
(i1,,3M-6)h ni) Generally, the mapping
of the stationary points of the ZPE -,
correlation energy-, and basis set error
corrected potential surface is E0 ?
E(HF-SCF/6-31G) Ecorr(REBECEP/6-31G)
ZPE(REZEP/6-31G) e.g. with Mulliken partial
charges. Comparison of the accuracy, time
(CPU) and disc space demand for different ZPE
approximations, namely the MMFF94,
HF-SCF/6-31G/freq and REZEP/6-31G methods on a
175 MHz Silicon Graphics machine see ref. 5.
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Efficiency of the REZEP method 4-5
Molecule Method ZPEkcal/mol Program
CPU time Disc-spaceMB Naphthalene MMFF94
92.44 Spartan
0.10s - HF-SCF/6-31G/freq 87.94 Gaussian98
1h-29m-15.0s 47 REZEP/6-31Ga
87.61 See REZEP eq. 1m-53.2s 20 a-iso-
cinchonine MMFF94 235.56 Spartan
0.97s - HF-SCF/6-31G/freq 227.41 Gauss
ian98 4d-21h-47m-19.7s 735 REZEP/6-31Gb 224.0
6 See REZEP eq. 1h-10m-58.5s 69 a. The
naphthalene (18 atoms and 68 electrons) was
included in the multi - linear fit (for REZEP
parameters). Its optimized B3LYP/6-31G
geometry was used. b. The a-iso-cinhonine (44
atoms and 158 electrons) was not included in the
fitting procedure. Its geometry was
optimized by MMFF94 molecular mechanics force
field.
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An approximation (M.E.Grice, P.Politzer
Chem.Phys.Lett. 244 (1995) 295) ZPE
kcal/mol ? 6.99nH 3.74nC 3.98nN 3.45nO
2.79nF 4.63 where nH is the
number of H atoms in the molecule, etc. 1.
Problems a. ZPE values of isomers are the same
! b. How to explain
the constant (M0 case) theoretically ? 2.
Compare to ZPE(REZEP)
? ? (A1,,M) EZPE(NA, ZA) in which we
weight with partial charges.
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Statistics 2 ZPE vs. N
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ZPE vs. N plots shows 6 no mathematical
correlation, but 0 lt ZPE lt b(N-1)
where 0 trivial value of b is about
0.0036 hartree lower envelope stiff
molecules, upper envelope b(N-1) hydrocarbons
(H atom has large REZEP atomic parameter),
middle part cyclic (H-deficient, aromatic)
compounds.
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Statistics 3 ZPE vs. sum of inverse atomic
masses
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ZPE vs. sum of inverse atomic masses shows 6
strong mathematical correlation
ZPE ? c S(A1,...,M) mA-1/2, where the
value of c is about 0.012 hartree, via linear
regression the constant is comparable to the
constant of Politzer equation for H atom
(mH 1) 0.01193 hartree mH-1/2 7.49
kcal/mol, it compares to the H-atom constant
in Politzer eq. (6.99 kcal/mol), etc.
(empirical constants have been explained
physically in this way). The classical
vibronic frequency ((2p)-1(kAAxx/mA)1/2)
algebraically conserved in the Hessian
(diagonal, cross derivative, ab initio values).
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An application catalytic enantioselective
hydrogenation of pyruvates
Adduct formation in liquid phase (hypothesis by
Margitfalvi and coworkers) before the
heterogeneous catalytic step. The DH(T0K)
values kcal/mol of the formation of
cinchonidine complexes (AB?AB) with methyl
pyruvates, acethon and 4-tertier
buthyl-ciclo-hexanon (TBCH). Complex HFSCF/6-
31G REBECEP /6-31G REBECEP /6-31G (closed
CD) (closed CD) (open CD) CD
Mepy 2.0 2.7 3.0 CD acetone 1.3 2.2 1.4
CD TBCH -0.3 0.7 0.4
See the details in other lectures.
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Another application CO oxidation on Pt/Sn (100
surface, 11 alloy), VASP program
VASP program periodic boundary condition,
pseudo-potential metal
bulk, metal surface, calculation of adsorption on
ab initio (DFT) level
CO on top part. charge free CO C 0.7457
O-0.7457 adsorbed CO Pt -0.2350 (on top)
Pt 0.0019 (another) C 1.1445 O
0.1514 The ability of cat. reaction can be
characterized with partial charges ...
PLAN Adsorption geometry and energy, vibronic
frequency in top, bridge, hollow position
mechanism of oxidation (on Pt/Sn the
reaction proceeds in much lower temperature
VASP ab initio / combined with
REBECEP-REZEP / combined with combi-chem !!!
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Conclusion
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References
1. Kristyan S. Chemical Physics, 224, 33-51
(1997)2. Kristyan S., Csonka G.I. Journal of
Computational Chemistry, 22, 241-254 (2001) 3.
Kristyan S., Csonka G.I. Theor. Chem. Acc., 106,
319-328 (2001)4. Ruzsinszky A., Kristyán S.,
Margitfalvi J.L., Csonka G.I. Journal of
Physical Chemistry A, 107, 1833-1839
(2003)5. Kristyan S., Pannon-8 Conference,
Szeged, Hungary, 2006 July in press6. Kristyan
S. J. Mol. Structure THEOCHEM, 712, 153-158
(2004)7. Kristyan S. Theor. Chem. Acc. (2005),
DOI 10.1007/s00214-005-0039-3, in pressThe
above publications can be downloaded in pdf form
from web.inc.bme.hu/kristyan