quantum calculations b. barbiellini [email protected] thematics seminar april 21,2005
TRANSCRIPT
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Goal: Solve the Schrödinger equation
Application: Description of chemical bonds
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Outline
• Independent Particle Approximation (IPM) and Hartree Fock (HF) SCF: Basis sets.
• Other theoretical methods: DFT and QMC.
• Illustrative example: Study of Hydrogen bond in ice and water.
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Electronic structure theoryH = E
Ab-initio - from the origins (First-principles)
No experimental parameters
Few physical constants c, h, me, qe
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min H| = E
Variational Theorem
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Theoretical Methods
• SCF & post-SCF methods (CI)
• Density functional theory (DFT)
• Stochastic methods: Quantum Monte Carlo (QMC)
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time
basis
set s
ize
me
tho
d
Climbing Mt. Psi
Correlation energy: energy contributions beyond SCF
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= det(r))det(r
Independent Particle Model:Hartree-Fock (HF) SCF
is a molecular orbitalis spin upF =e F is an effective one-particle hamiltonian which depend on MO’s Self Consistent Field (SCF).
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• Linear combination of atomic orbitals termed “basis functions”
Basis set – mathematical representation of molecular orbitals
• Minimal basis set – one basis function for every atomic orbital that is required to describe the free atom
H(1s) C(1s,2s,2p) → CH4: 9 basis functions• Larger basis sets are more flexible
– better approximation of exact MOs• Polarization functions, diffuse functions
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• Slater-type orbitals (J.C. Slater)
– Represent electron density well in valence region and beyond (not so well near nucleus)
– Evaluating these integrals is difficult
• Gaussian-type orbitals (F. Boys)
– Easier to evaluate integrals, but do not represent electron density well
– Overcome this by using linear combination of GTOs
STOs v. GTOs
g ,r cx n y m z le r 2
d
pg
pp
s( ,r ) cx n y m z le r
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Density functional theory
• Less expensive than post-SCF methods
• Include some electron correlation
• Eelec = ET + EV + EJ + EXC
• Pure functionals: BP86, BLYP
• Hybrid HF/DFT: B3LYP
• Good for geometries, electron affinities
• Good for large systems
• Problem: not systematic
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Example:Gaussian Input
#RHF/6-31G(d) Pop=Full Test
RHF/6-31G(d) formaldehyde single point
0,1C 0.0 0.0 0.0O 0.0 1.22 0.0 H 0.94 -0.54 0.0H -0.94 -0.54 0.0
method basis set key words
} route sectionblank line
blank line} title section
charge, multiplicity
}molecular structure section atomic symbols (or numbers) xyz coordinates (or z-matrix)
blank line
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Quantum Monte Carlo
• Deals with the many body wave-function.
• Include electron correlation (Jastrow terms).
• Variation QMC --- Stochastic Gradient Approximation (SGA).
• Diffusion QMC (almost exact, fixed node approximation) --- computational expensive.
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Distance H-H
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Scattered x rays in iceIsaacs et al., PRL 82 (1999) 600
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Wavelike fringes corresponding to interference between the electrons on neighboring sigma and hydrogen bonding sites
Compton Profile Anisotropy
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B(r) Fourier transform CP: MO orbital autocorrelation function
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Conclusion
Quantum calculations are of interest because they can deal with electronic effects, electron de-localization, charge-transfer, and other phenomena, which are otherwise difficult or impossible to treat at the level of classical mechanics.
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