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NANO266 - Lecture 2 - The Hartree-Fock Approach

UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.

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The Hartree-Fock
Approximation
Shyue Ping Ong
Stationary Schrödinger Equation for a System of
Atoms
where
NANO266
2
Eψ = Hψ
H = −
h 2
2me
∇i
2
i
∑ −
h 2
2mk
∇k
2
−
e2
Zk
rikk
∑
i
∑ +
e2
rijj
∑
i
∑
k
∑ +
Zk Zle2
rkll
∑
k
∑
KE of electrons
KE of nuclei
Coulumbic attraction
between nuclei and
electrons
Coulombic repulsion
between electrons
Coulombic repulsion
between nuclei
Stationary Schrödinger Equation inAtomic Units
To simplify the equations a little, let us from
henceforth work with atomic units
NANO266
3
Dimension Unit Name Unit Symbol
Mass Electron rest mass me
Charge Elementary Charge e
Action Reduced Planck’s constant ħ
Electric constant Coulomb force constant ke
H = −
1
2
∇i
2
i
∑ −
1
2mk
∇k
2
−
Zk
rikk
∑
i
∑ +
1
rijj
∑
i
∑
k
∑ +
Zk Zl
rkll
∑
k
∑
TheVariational Principle
We can judge the quality of the wave functions by
the energy – the lower the energy, the better. We
may also use any arbitrary basis set to expand
the guess wave function.
How do we actually use this?
NANO266
4
φHφ dr∫
φ2
dr∫
≥ E0
Linear combination of atomic orbitals (LCAO)
NANO266
5
http://www.orbitals.com/
Solving the one-electron molecular system with
the LCAO basis set approach
In general, we may express our trial wave functions
as a series of mathematical functions, known as a
basis set.
For a single nucleus, the eigenfunctions are
effectively the hydrogenic atomic orbitals. We may
use these atomic orbitals as a basis set for our
molecular orbitals. This is known as the linear
combination of atomic orbitals (LCAO) approach.
NANO266
6
φ = aiϕi
i=1
N
∑

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NANO266 - Lecture 2 - The Hartree-Fock Approach

  • 2. Stationary Schrödinger Equation for a System of Atoms where NANO266 2 Eψ = Hψ H = − h 2 2me ∇i 2 i ∑ − h 2 2mk ∇k 2 − e2 Zk rikk ∑ i ∑ + e2 rijj ∑ i ∑ k ∑ + Zk Zle2 rkll ∑ k ∑ KE of electrons KE of nuclei Coulumbic attraction between nuclei and electrons Coulombic repulsion between electrons Coulombic repulsion between nuclei
  • 3. Stationary Schrödinger Equation inAtomic Units To simplify the equations a little, let us from henceforth work with atomic units NANO266 3 Dimension Unit Name Unit Symbol Mass Electron rest mass me Charge Elementary Charge e Action Reduced Planck’s constant ħ Electric constant Coulomb force constant ke H = − 1 2 ∇i 2 i ∑ − 1 2mk ∇k 2 − Zk rikk ∑ i ∑ + 1 rijj ∑ i ∑ k ∑ + Zk Zl rkll ∑ k ∑
  • 4. TheVariational Principle We can judge the quality of the wave functions by the energy – the lower the energy, the better. We may also use any arbitrary basis set to expand the guess wave function. How do we actually use this? NANO266 4 φHφ dr∫ φ2 dr∫ ≥ E0
  • 5. Linear combination of atomic orbitals (LCAO) NANO266 5 http://www.orbitals.com/
  • 6. Solving the one-electron molecular system with the LCAO basis set approach In general, we may express our trial wave functions as a series of mathematical functions, known as a basis set. For a single nucleus, the eigenfunctions are effectively the hydrogenic atomic orbitals. We may use these atomic orbitals as a basis set for our molecular orbitals. This is known as the linear combination of atomic orbitals (LCAO) approach. NANO266 6 φ = aiϕi i=1 N ∑
  • 7. The Secular Equation NANO266 7 E = aiϕi i=1 N ∑ " # $ % & 'H aiϕi i=1 N ∑ " # $ % & 'dr∫ aiϕi i=1 N ∑ " # $ % & ' 2 dr∫ = aiaj ϕiHϕj dr∫ ij ∑ aiaj ϕiϕj dr∫ ij ∑ = aiaj Hij ij ∑ aiajSij ij ∑ Resonance integral Overlap integral
  • 8. The Secular Equation,contd To minimize the energy, Which gives Or in matrix form NANO266 8 ∂E ∂ak = 0, ∀k ai (Hki − ESki ) i=1 N ∑ = 0, ∀k H11 − ES11 H12 − ES12 ! H1N − ES1N H21 − ES21 H22 − ES22 ! H2N − ES2N " " # " HN1 − ESN1 HN 2 − ESN 2 ! HNN − ESNN " # $ $ $ $ $ % & ' ' ' ' ' a1 a2 " aN " # $ $ $ $ $ % & ' ' ' ' ' = 0
  • 9. The Secular Equation,contd Solutions exist only if Procedure: i.  Select a set of N basis functions. ii.  Determine all N2 values of Hij and Sij. iii.  Form the secular determinant and determine the N roots Ej. iv.  For each Ej, solve for coefficients ai. NANO266 9 H11 − ES11 H12 − ES12 ! H1N − ES1N H21 − ES21 H22 − ES22 ! H2N − ES2N " " # " HN1 − ESN1 HN 2 − ESN 2 ! HNN − ESNN = 0
  • 10. HückelTheory Basis set formed from parallel C 2p orbitals Overlap matrix is given by Hii = Ionization potential of methyl radical Hij for nearest neighbors obtained from exp and 0 elsewhere NANO266 10 Sij =δij
  • 11. The Born-OppenheimerApproximation Heavier nuclei moves much more slowly than electrons => Electronic relaxation is “instantaneous” with respect to nuclear motion Electronic Schrödinger Equation NANO266 11 (Hel +VN )ψel (qi;qk ) = Eelψel (qi;qk ) Electronic energy Constant for a set of nuclear coordinates
  • 12. Stationary Electronic Schrödinger Equation where NANO266 12 Eelψel = Helψel Hel = − 1 2 ∇i 2 i ∑ − Zk rikk ∑ i ∑ + 1 rijj ∑ i ∑ KE and nuclear attraction terms are separable H = hi i ∑ where hi = − 1 2 ∇i − Zk rikk ∑
  • 13. Hartree-ProductWave Functions Eigen functions of the one-electron Hamiltonian is given by Because the Hamiltonian is separable, NANO266 13 hiψi =εiψi ψHP = ψi i ∏ HψHP = hi i ∑ ψk k ∏ = εi i ∑ # $ % & ' (ψHP
  • 14. The effective potential approach To include electron-electron repulsion, we use a mean field approach, i.e., each electron sees an “effective” potential from the other electrons NANO266 14 hi = − 1 2 ∇i − Zk rikk ∑ +Vi, j where Vi, j = ρj rij ∫ j≠i ∑ dr
  • 15. Hartree’s Self-Consistent Field (SCF)Approach NANO266 15 Guess MOs Construct one- electron operations hi Solve for new ψ hiψi =εiψi Iterate until energy eigenvalues converge to a desired level of accuracy E = εi i ∑ − 1 2 ψi 2 ψj 2 rij dri drj∫∫ What’s the purpose of this term?
  • 16. What about the Pauli Exclusion Principle? Two identical fermions (spin ½ particles) cannot occupy the same quantum state simultaneously è Wave function has to be anti-symmetric For two electron system, we have NANO266 16 ψSD = 1 2 ψa (1)α(1)ψb (2)α(2)−ψa (2)α(2)ψb (1)α(1)[ ] = 1 2 ψa (1)α(1) ψb (1)α(1) ψa (2)α(2) ψb (2)α(2) where α is the electron spin eigenfunction Slater determinant
  • 17. For many electrons… NANO266 17 ψSD = 1 N! χ1(1) χ2 (1) ! χN (1) χ1(2) χ2 (2) ! χN (2) ! ! " ! χ1(N) χ2 (N) ! χN (N) where χk are the spin orbitals
  • 18. The Hartree-Fock (HF) Self-Consistent Field (SCF) Method NANO266 18 fi = − 1 2 ∇i 2 − Zk rik +Vi HF {j} k nuclei ∑ F11 − ES11 F12 − ES12 ! F1N − ES1N F21 − ES21 F22 − ES22 ! F21 − ES2N " " # " FN1 − ESN1 FN 2 − ESN 2 ! FNN − ESNN = 0 HF Secular Equation Fµυ = µ |− 1 2 ∇i 2 |υ − Zk µ | 1 rk |υ + Pλσ λσ ∑ (µυ | λσ )− 1 2 (µλ |υσ ) $ %& ' () k nuclei ∑ Weighting of four-index integrals by density matrix, P
  • 19. Flowchart of HF SCF Procedure NANO266 19
  • 20. Limitations of HF Fock operators are one-electron => All electron correlation, other than exchange, is ignored Four-index integrals leads to N4 scaling with respect to basis set size NANO266 20 Ecorr = Eexact − EHF
  • 21. PracticalAspects of HF Calculations Basis Sets Effective Core Potentials Open-shell vs Closed- shell Accuracy Performance NANO266 21
  • 22. Basis Set Set of mathematical functions used to construct the wave function. In theory, HF limit is achieved by an infinite basis set. In practice, use finite basis sets that can approach HF limit as efficiently as possible NANO266 22
  • 23. Contracted Gaussian Functions Slater-type orbitals (STO) with radial decay cannot be analytically integrated -> Use linear combination of Gaussian-type orbitals (GTOs) with radial decay to approximate STOs STO-3G •  STO approximated by 3 GTOs •  Known as single-ζ or minimal basis set. NANO266 23 e−r2 e−r
  • 24. Multiple-ζ and Split-Valence Multiple-ζ •  Adding more basis functions per atomic orbital •  Examples: cc-pCVDZ, cc-pCVTZ (correlation-consistent polarized Core and Valence (Double/Triple/etc.) Zeta) Split-valence or Valence-Multiple-ζ •  Still represent core orbitals with single, contracted basis functions •  Valence orbitals are split into many functions (Why?) •  Examples: 3-21G, 6-31G, 6-311G NANO266 24 # of primitives in core # of primitives in valence
  • 25. Polarization and Diffuse Functions Polarization functions •  Description of MOs require more flexibility than provided by AOs, e.g., NH3 is predicted to be planar if using just s and p functions •  Additional basis functions of one quantum number of higher angular momentum than valence, e.g., first row -> d orbitals •  Notation: 6-31G* [old] or 6-31G(d) [new], 6-31(2d,p) [2d functions for heavy atoms, additional p for H] Diffuse functions •  Highest energy MOs of anions, highly excited states tend to be more diffuse •  Augment standard basis sets with diffuse functions •  Notation: 6-31+G, 6-311++G(3df, 2pd), aug-cc-pCVDZ NANO266 25
  • 26. Effective Core Potentials Heavy atoms have many electrons •  Intractable to model all of them, even with a minimal basis set •  However, most of the electrons are in the core Solution: Replace core electrons with analytical functions (effective core potentials or ECPs) that represent combined nuclear-electronic core to the remaining electrons Key selection decision: How many electrons to include in the core? NANO266 26
  • 27. Open-shell vs closed-shell Restricted HF (RHF) •  Closed-shell systems, i.e., no unpaired electrons Restricted open-shell HF (ROHF) •  Use RHF formalism, but with density matrix for singly occupied orbitals not multiplied by a factor of 2. •  Wave functions are eigenfunctions of S2 •  But fails to account for spin polarization in doubly occupied orbitals Unrestricted HF (UHF) •  Includes spin polarization •  Wave functions are not eigenfunctions of S2, i.e., spin contamination NANO266 27
  • 28. Accuracy Energetics •  In general, extremely poor; correlation is extremely important in chemical bonding! •  Protonation energies are typically ok (no electrons in H+) •  Koopman’s Theorem: First IE is equal to the negative of the orbital energy of the HOMO Geometry •  Typically relatively good ground state structures with basis sets of modest size •  But transition states (with partial bonding) can be problematic, as well as some pathological systems NANO266 28
  • 29. Performance Formal N4 scaling But in reality, speedups can be achieved through: •  Symmetry •  Estimating upper bounds to four-index integrals •  Fast multipole and linear exchange integral computations For practical geometry optimizations, frequently helps to first compute geometry with a smaller basis set to provide a better initial geometry and a guess for the Hessian matrix. NANO266 29