Quantum Calculations

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# Quantum Calculations - PowerPoint PPT Presentation

Quantum Calculations. B. Barbiellini bba@neu.edu Thematics seminar April 21,2005. Goal: Solve the Schrödinger equation . Application: Description of chemical bonds. Outline. Independent Particle Approximation (IPM) and Hartree Fock (HF) SCF: Basis sets.

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### Quantum Calculations

B. Barbiellini bba@neu.edu

Thematics seminar

April 21,2005

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

VariationalTheorem

min<| H|> = E

Theoretical Methods

• SCF & post-SCF methods (CI)
• Density functional theory (DFT)
• Stochastic methods: Quantum Monte Carlo (QMC)

Climbing Mt. Psi

Correlation energy: energy contributions beyond SCF

Independent Particle Model:

Hartree-Fock (HF) SCF

 = det(j(a,r))det(j(b,r))

• is a molecular orbital

a is spin up

F j =e j F is an effective one-particle hamiltonian which depend on MO’s  Self Consistent Field (SCF).

Larger basis sets are more flexible

• better approximation of exact MOs
• Polarization functions, diffuse functions

Basis set – mathematical representation of molecular orbitals

• Linear combination of atomic orbitals termed “basis functions”
• 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

STOs v. GTOs

• 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

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

basis set

method

key words

} route section

blank line

blank line

charge, multiplicity

}

molecular structure section

atomic symbols (or numbers)

xyz coordinates (or z-matrix)

blank line

#RHF/6-31G(d) Pop=Full Test

RHF/6-31G(d) formaldehyde single point

0,1

C 0.0 0.0 0.0

O 0.0 1.22 0.0

H 0.94 -0.54 0.0

H -0.94 -0.54 0.0

} title section

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.

b

Distance H-H

Scattered x rays in ice

Isaacs et al., PRL 82 (1999) 600

Compton Profile Anisotropy

Wavelike fringes corresponding to interference between the electrons on neighboring sigma and hydrogen bonding sites

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.