Beyond the HF Approximation

Shyue Ping Ong

Overview

In this lecture, we will only touch on conceptual underpinnings of how correlation is included, without going too much into the details of the methods. Most of the advanced methods are far too computationally expensive and limited to small system sizes, which makes them less useful for the materials scientist at this time. It suffices that you understand them at a conceptual level, and if you are interested (or they become more accessible in future), there are many excellent works on the subject. NANO266 2

Limitations of HF

All correlation, other than exchange, is ignored in HF

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Ecorr = Eexact −EHF

So how might one improve on HF?

HF utilizes a single determinant. An obvious extension is Types of correlation

•  Dynamic correlation: From ignoring dynamic electron-electron interactions. Typically c0 is much larger than other coefficients.

•  Non-dynamical correlation: Arises from single determinant nature of HF. Several ci with similar magnitude as c0.

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ψ = c0ψHF + c1ψ1 + c2ψ2 +…

Degenerate frontier orbitals cannot be represented with single determinant!

Multiconfiguration SCF

Optimize orbitals for a combination of configurations (orbital occupations)

•  Configuration state function (CSF): molecular spin state and occupation number of orbitals

•  Active space: orbitals that are allowed to be partially occupied (based on chemistry of interest)

Scaling CAS: Complete active space (CASSCF)

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# of singlet CSFs for m electrons in n orbitals = n!(n+1)!m2

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Full Configuration Interaction (CI)

CASSCF calculation of all orbitals and all electrons Best possible calculation within limits of basis set

For small systems, can be used to benchmark other methods

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Full CI Infinite Basis Set

Exact solution to Schodinger Equation

Limiting excitations in CI

CIS (CI singles) •  Used for excited

states •  No use for ground

states

CID (CI doubles) CISD (CI singles doubles)

•  N6 scaling

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Møller–Plesset perturbation theory

Treats exact Hamiltonian as a perturbation on sum of one-electron Fock operators

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H = H (0) +λV = fii∑ +λV

Expanding the ground-state eigenfunctions and eigenvalues as Taylor series in λ,ψ =ψ (0) +λψ (1) +λ 2ψ (2) +…

a = a(0) +λa(1) +λ 2a(2) +…

where ψ (k ) =1k!∂kψ∂λ k and a(k ) =

1k!∂ka∂λ k

H ψ = a ψ

∴(H (0) +λV ) λ kψ (k )∑ = λ ka(k ) λ kψ (k )∑∑By equating powers of λ and imposing normalization, we can derivea(k ), which are the k th order corrections to a(0).

MPn Theory

MP1 is simply HF MP2

•  Second-order energy correction •  Analytic gradients available •  N5 scaling

MPn > 2

•  No analytic gradients available •  > 95% of electron correlation at n=4.

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Issues in Perturbation Approach

Perturbation theory works best when perturbation is small (convergence of Taylor series expansion)

•  In MPn, perturbation is full electron-electron repulsion!

MPn is not variational! (possible for correlation to be larger than exact, but in practice, basis set limitations cause errors in opposite direction)

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Coupled-Cluster

Full-CI wave function can be described as

If we truncate at T2

CCSD(T) •  Includes single/triples coupling term •  Analytic gradients and second derivatives available •  Gold-standard in most quantum chemistry calculations

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ψ = eTψHF

where T = T1 +T2 +T3 +…+Tn is the cluster operator

ψCCSD = (1+ (T1 +T2 )+(T1 +T2 )

2

2!+…)ψHF

Practical Considerations

Basis set convergence is a bigger problem for correlated calculations

Performance vs Accuracy •  HF < MP2 ~ MP3 < CCD < CISD < QCISD ~CCSD < MP4 < QCISD(T) ~ CCSD(T)

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Relative accuracy of variational methods – Dissociation of HF (hydrogen fluoride)

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Foresman, J. B.; Frisch, Ae. Exploring Chemistry With Electronic Structure Methods: A Guide to Using Gaussian; Gaussian, 1996.

Ionization potentials

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Errors introduced via the truncation of the space at different excitation levels and the effect of this on the IP. The two systems are oxygen in an aug-cc-pVQZ basis and neon in an aug-cc-pVTZ basis set. The dashed lines indicate the difference in the total energy of each species compared to the FCI limit, and the solid lines indicate the error in the IP with each species truncated at the given excitation level.

J. Chem. Phys. 132, 174104 (2010); http://dx.doi.org/10.1063/1.3407895

Relative computational cost – C5H12

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Foresman, J. B.; Frisch, Ae. Exploring Chemistry With Electronic Structure Methods: A Guide to Using Gaussian; Gaussian, 1996.

Bond lengths

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Cramer, C. J. Essentials of Computational Chemistry: Theories and Models; 2004.

Parameterized methods

G2/G3 theory for accurate thermochemistry (errors < 4 kcal / mol)

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References

Essentials of Computational Chemistry: Theories and Models by Christopher J. Cramer

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