quantum calculations b. barbiellini [email protected] thematics seminar april 21,2005
TRANSCRIPT
Goal: Solve the Schrödinger equation
Application: Description of chemical bonds
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.
Electronic structure theoryH = E
Ab-initio - from the origins (First-principles)
No experimental parameters
Few physical constants c, h, me, qe
min H| = E
Variational Theorem
Theoretical Methods
• SCF & post-SCF methods (CI)
• Density functional theory (DFT)
• Stochastic methods: Quantum Monte Carlo (QMC)
time
basis
set s
ize
me
tho
d
Climbing Mt. Psi
Correlation energy: energy contributions beyond SCF
= 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).
• 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
• 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
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
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
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charge, multiplicity
}molecular structure section atomic symbols (or numbers) xyz coordinates (or z-matrix)
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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.
Distance H-H
Scattered x rays in iceIsaacs et al., PRL 82 (1999) 600
Wavelike fringes corresponding to interference between the electrons on neighboring sigma and hydrogen bonding sites
Compton Profile Anisotropy
B(r) Fourier transform CP: MO orbital autocorrelation function
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.
