International Workshop on Computational Methods

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International Workshop on Computational
Methods For Nanoscale Systems
XMVB An ab initio Nonorthogonal Valence Bond
Program
Wei Wu
Department of Chemistry, Xiamen University
Dec. 11-13, 2006 Hong Kong
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Quantum Chemistry
Molecular Orbital Theory
Valence Bond Theory
Delocalized MO based
Localized AO based
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Roots of Valence Bond Theory
1916, G.N. Lewis, J. Am. Chem. Soc. 38, 762
(1916). The Atom and the
Molecule.
Introduces the notions of electron-pair bonding
and octet rule, and forms the basis for
subsequent VB theory.
1927, Heitler and London, Zeits. für Physik. 44,
55(1927). Interaction Between Neutral Atoms and
Homopolar
Bonding in H2 can be accounted for by wave
function of quantum mechanics. (Leitler-London
function)
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1929, Slater presented his determinant method
Phys. Rev. 34, 1293 (1929).
1931, Slater generalized HL method to n-electron
systems
Phys. Rev. 38, 1109 (1931).
1931, Pauling constructed a general quantum
chemical theory for polyatomic systems, and
Linked quantum mechanics to the chemical theory
of Lewis.
J. Am. Chem. Soc. 53, 1367(1931) 3225(1931).
VB wave function Heitler-London-Slater-Pauling(HL
SP) function.
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Origins of MO Theory
R.S.Mulliken, Phys. Rev. 32, 186(1928)716(1928)
33, 730(1929)41,49(1932).
F.Hund, Zerta. für Physik, 51, 759(1928)
73,1(1931).
They made spectral and quantum number
assignments of electrons in molecules, based on
correlation diagrams.
They developed an alternative approach called
Molecular Orbital Theory.
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VB MO Rivalry
VB and MO theories have competed since
1930s, which seems to accompany chemistry to the
21th Century.
This rivalry involved most of the prominent
chemists of various periods. (Mulliken, Huckel,
J. Mayer, Robinson, Lapworth, Ingold, Sidgwick,
Lucas, Barlett, Dewar, Longuet-Higgins, Coulson,
Roberts, Winstein, Brown, and so on)
S.G.Brush. Stud. Hist. Phil.Sci. 30, 21263(1999).
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By the end of World War II, VB resonance
theory was widely applied to all known structural
chemistry (in Paulings book) and to a variety of
problems in organic chemistry (in Whelands
book). VB-resonance theory dominated
chemists thinking.
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• Since the mid 1950, many successes of MO theory
• The experimental verification of Hückel Rules
• The construction of intuitive MO theory and
  their wide applicability for rationalization of
  structures and spectra
• The highly successful predictive applications
  of MO theory in chemical reactivity
• The development of MO-based computational
  techniques (Extended Hückel and semiempirical
  programs).

VB theory has ceased to guide experimental
chemists to new experiments.
Complete victory of MO theory!
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Though MO theory dominates quantum chemistry
calculations, Chemists continue to use the
language of VB theory and the associated concepts
of Lewis structures, local bonds, hybridization,
curved arrows, resonance and so on, to
rationalize chemical reactivity.
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Advantages of Valence Bond Theory

• describes correctly a process of bond forming /
  bond breaking with a very compact wave function,
  providing a very simple and intuitive tool for
  the understandings of chemical bonds.
• calculates the energy of individual VB
  structures (diabatic states), which provides
  chemical insights into the origin of reaction
  barriers.
• calculates the resonance energy, which is an
  important concept in electronic delocalization
  effects.

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Methodology
In VB theory, a many-electron wave
function is expressed in terms of VB functions.
corresponds to a given VB structure
H2
H H H- H H H-
is given by solving secular equation
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Heitler-London Function
H2 H H H- H
H H-
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C6H6
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General Cases
A VB function for a system with m covalent
bonds is expressed in terms of 2m determinants.
For matrix element 22m determinants
C2H6, N 14, n 7, 128 determinants for a VB
function 16384
determinants for a matrix element
Difficulties in VB theory

• Multi-determinants for a VB function
• Non-orthogonal atomic orbitals

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Spin-free VB function
In spin-free quantum chemistry, matrix
elements of a spin independent Hamiltonian are
determined only by the spin-free function.
where
is the standard projection operator as
?K is a product of orbitals
is the spin-free form of VB function
irreducible representation matrix element.
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Matrix elements of the Hamiltonian and overlap
Non-orthogonal
a and b depend on permutation P
or
includes N! terms.
N! problem in VB theory
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Paired-permanent-determinant Algorithm (PPD)
W.Wu et al. IJQC,67(1998), 287.
PPD function
matrix
Determinant
Permanent
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Evaluation of PPDs
A PPD of order N can be expressed in terms of
N(N-1)/2 products of PPDs of order N-2 and their
corresponding minors, which is similar to the
Laplace expansion for determinants.
if
and
but
if
if
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Example N 4, S 0
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N 4, S 1 (1 and 2 are PSI, but 3 and 4 are
ASI)
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• For even N, S 0, we have

,
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Hamiltonian and overlap matrix elements
A VB function can be expressed as a PPD function
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Overlap Matrix Element
An overlap matrix element is also a PPD function
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Hamiltonian Matrix Elements S 0
.
.
,
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Obviously, computing a PPD is much more
expensive than that of a determinant, and thus,
the PPD algorithm is not a final solution to N!
problem, but it is more suitable to systems with
many covalent bonds.
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Classical VB method
Uses covalent and ionic structures based on
unoptimized atomic orbitals.
FF
F F F F
Extremely poor.
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Valence Bond Self-consistent Field (VBSCF) Both
structure coefficients and VB orbitals are
optimized simultaneously to minimize the total
energy.
FF
F F F F
provides qualitative correct description for bond
breaking/forming, but its accuracy is still
wanting. VBSCF takes care of the static
correlation, but lacks dynamic correlation.
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Breathing Orbital Valence Bond (BOVB)

• Different orbitals for different VB structures

FF
F F F F
The degree of freedom introduces dynamic
correlation with very compact wave function. It
improves considerably the accuracy of the results.
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Valence Bond Configuration Interaction (VBCI)
      (a) Wu, W et al. J. Phys. Chem. A 2002,
105, 2721. (b) Song, L et al. J. Comput.
Chem. 2004, 25, 472.
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Xiamen Valence Bond
XMVB An ab initio Nonorthogonal Valence Bond
Program
Version 1.0 Lingchun Song, Yirong Mo, Qianer
Zhang, Wei Wu   Center for Theoretical
Chemistry, State Key Laboratory for Physical
Chemistry of Solid Surfaces, and Department of
Chemistry, Xiamen University, Xiamen, Fujian
361005, CHINA weiwu_at_xmu.edu.cn
J. Comp. Chem. 26, 514-521 (2005).
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VB methods implemented in XMVB

• Hartree-Fock Method
• VBSCF
• BOVB
• VBCI
• Total Energy, Energy for individual Structure,
  Dipole
• Moments, Weights,
• Plot VB orbitals
• Parallelization (MPI)

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Comparison with Other Valence Bond Codes
Goddard GVB/GAUSSIAN, Cooper et al
SCVB/Molpro Van Lenthe et al Turtle/GAMESS-UK Wei
Wu et al XMVB Li Jiabo et al VB2000 Gallup et
al CRUNCH
Pure VB methods Strictly localized AOs are
used, HLSP functions are used
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Dr. Braidas mail (Nov.20, 2006) I've asked
Philippe as well, and actually we both didn't use
TURTLE for some time, several years in fact, for
the following reasons- XMVB is more
user-friendly,- you did improve XMVB very
efficiently not long ago (from Xiamen-99  to
XMVB), and this version is much faster than
TURTLE when we do  BOVB, especially when we do
SD-BOVB (and the latest version of XMVB seems to
be again a step forward), .
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P. C. Hibertys comments
XIAMEN program is far easier to use than TURTLE.
XIAMEN program converges better, is less
sensitive to the choice of the guess, can find a
good guess automatically and has an automatic
BOVB option.
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XMVB Applications
Hydrogen transfer J. Phys. Chem. A., 105, 8226
(2001). J. Phys. Chem. A, 106, 11361 (2002).
Chem. Eur. J. 9, 4540 (2003). J. Am. Chem. Soc.
126, 13539(2004). J. Am. Chem. Soc. 128, 2836
(2006). Charge transfer J. Am. Chem. Soc.,
126,3974(2004). Proton transfer J. Am. Chem.
Soc., 122, 6754 (2000). J. Phys. Org. Chem. 12,
259 (1999). J. Phys. Chem. 106, 11616 (2002).
Can. J. Chem., 83, 1649(2005) SN2 reaction
Chem. Eur. J., 12 7458(2006). Bonding and
electronic delocalization Angew. Chem. Int. Ed.
43, 1986(2004). J. Phys. Chem. 107, 10011(2003).
J. Am. Chem. Soc., 124, 11781 (2002). J. Phys.
Chem. A., 109, 2310 (2005) J. Chem. Theo.
Comp., 1, 394(2005). J. Comp. Chem.,
28,185(2006).
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The Origin of Rotation Barrier in Ethane
Mo, Wu, et al. Angew. Chem. Int. Ed. 43, 1986
(2004).
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Origin of Barrier?
Steric Repulsion Model
Hyperconjugation Model
F. Weinhold, J. Am. Chem. Soc. 1979, 101, 1700
Angew. Chem. Int. Ed. 2003, 42, 4188.
L. Pauling, The Natrue of Chemical Bond, 3rd,
1960 M. Karplus, J. Chem. Phys. 1968, 49, 2592.
L. Goodman, Nature, 2001, 411, 565.
E. J. Baerends, Angew. Chem. Int. Ed., 2003, 42,
4183.
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(No Transcript)
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MO Method Delocalized MOs
optimize
Minimum of Energy
Orbtial transformation
Localized MOs
Lower energy?
optimize
PossibilityOverestimate delocalization energy ?
VB method Localized AOs
Minimum of Energy
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Ab initio VB 14e/7 bonds/6-311G
PPD algorithm is used.
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Energy analyses with the ab initio VB method and
6-31G(d)
The hyperconjugation effect does favor the
staggered structure but accounts for only around
one-third of the rotation barrier, most of which
comes from the steric hindrance.
Mo, Wu, et al. Angew. Chem. Int. Ed. 43, 1986
(2004).
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Figure 1.Comparison of energy profiles (energy E
versus dihedral angle f) for the ethane rotation.
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Hybrid Methods with VB Theory
VB-DFT method
Chem. Phys. Lett., 301(1999) 37-42.
VBPCM method
J. Phys. Chem. A, 108(2004) 6017-6024.
VB/MM method
A. Shurki, HA. Crown, J. Phys. Chem. B,
109(2005), 23638.
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Summary

• As complement to the MO methods, VB method is
  able to provide intuitive insights into chemical
  problems.
• Though VB calculation is still expensive, but
  its cost is acceptable, compared to post-HF
  methods.
• Compared to other VB programs, XMVB package is
  more efficient and more user-friendly, and
  provides an ideal tool for VB applications.

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Acknowledgements
Dr. Lingchun Song (Xiamen University,
Xiamen) Prof. S. Shaik (Hebrew University,
Jerusalem) Prof. P. C. Hiberty (University of
Paris-11, Paris) Prof. Y. Mo (Western Michigan
University, Kalamazoo) Natural Science
Foundation of China Ministry of Education,
China Ministry of Science Technology, China
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Thanks !