perturbatively added on top of CCSD, has been named the gold standard of chemistry as it generally yields highly accurate results with an expensive, but an acceptable scaling
of O(N7
bas), where the most costly O(Nbas7 ) is not iterative. Recent improvements [114]
within the framework of pair-natural orbital approaches, has brought down the apparent scaling of CCSD(T) to linear, allowing to compute the energies of complete proteins on the level of CCSD(T).
4.5.5
Excited states methods
In most of our discussion up to this point we have only focused on obtaining an approxim- ation to the ground state of the electronic Schrödinger equation. In some applications of electronic structure theory, however, electronic excitations play a role. Examples include the interaction of UV photons or photons of visible light with the electronic structure in a dye or a solar cell or more generally any photo-activated chemical reaction. Whenever this is the case the modelling of multiple electronic states on an equal footing is required.
For FCI or the truncated CI methods, this can be achieved without additional modification by solving the respective full or truncated CI matrix for more than one eigenpair. All but the lowest are excited states. These are not the only excited states methods in existence. In fact to each of the other methods we have discussed so far one is able to appoint at least one analogue [115]. For example for Hartree-Fock, there is configuration-interaction singles (CIS) or time-dependent HF (TDHF) and for coupled- cluster there are the equation-of-motion and linear-response coupled-cluster theories [116, 117]. Last but not least, the algebraic-diagrammatic construction scheme (ADC) for the polarisation propagator at various orders [118, 119] can be seen as a CI-like scheme on top of a Møller-Plesset ground state. Its excited states are generally in good agreement with the MP description of the ground state.
4.6
Density-functional theory
In this section we want to briefly look at a different approach towards modelling the
electronic structure. Instead of solving for the wave function Ψ0associated to the ground
state of the electronic Hamiltonian ˆHNelec, the idea behind density-functional theory
is to solve for the state’s electronic density ρ0 instead.
The rationale for this is twofold. Firstly the density contains all information about the
chemical system. The integralRR3ρ(r) dr evaluates to the number of electrons Nelec and
via Kato’s cusp condition [69] one may obtain the nuclear charges ZA via the derivatives
of the electron density at the cusp points. Secondly the Hohnberg-Kohn theorems [120] as well as the Levy constrained search ansatz [121] provide a unique identification between a particular ground state electron density and the potential, which generates this density.
Even from a mathematical point of view solving for the ground state density ρ0(r) is
thus sufficient to characterise all properties of the ground state of a system.
The Levy constrained search ansatz [121] provides a conceptionally rather intuitive route to obtain the ground state density, namely by a constrained minimisation of the energy with respect to all possible densities. The issue with this procedure is that a closed-form expression for the energy functional E(ρ), which returns the energy of a given
82 CHAPTER 4. SOLVING THE MANY-BODY ELEC. SCHRÖDINGER EQN. density, is not known for any relevant chemical system. In other words Levy constrained search in the form presented so far cannot be applied to chemical systems.
Further progress can be made with the Kohn-Sham ansatz [122], however. The idea
is to consider a fictitious system of Nelec non-interacting electrons, which still has the
property that it reproduces the exact ground state density of the full, interacting system. Ignoring spin in our discussion in this model system the exact wave function is a single determinant Ψ = ΦΘ= N^elec i=1 ψi where Θ ≡ (ψ1, ψ2, . . . , ψNelec) ∈ H 1(R3, C)Nelec
is a tuple of Nelec single-particle functions. Ignoring spin the resulting ground state
density is
ρΘ(r) =
NXelec
i=1
|ψi(r)|2, which allows to write the Kohn-Sham energy functional as
EKS(Θ) = 1 2 NXelec i=1 Z R3k∇ψik 2 2dr + Z R3 M X A=1 ZAρΘ(r) kr − RAk2 dr +1 2 Z R3 Z R3 ρΘ(r1)ρΘ(r2) kr1− r2k2 dr1dr2+ Exc(ρΘ). (4.98)
In this expression Exc is the exchange-correlation functional, which depends only
on the density function ρ. This term is supposed to describe the non-local many-body effects not yet contained in the other terms, which is threefold, (1) the part of the kinetic energy missed by the non-interacting electrons, (2) the exchange interaction as well as (3) correlation effects. The crux with Kohn-Sham DFT is that its exact functional form is unknown, such that one has to live with approximations. Which exchange-correlation functional is sensible for a particular problem depends very much on the context of the chemical system, the property one is interested in and is still subject of debate in quantum-chemical literature. Notice, however, that if the exact exchange-correlation functional was to be found, (4.98) would yield the exact ground-state energy.
Following the original Levy constrained search, we want to find the density corres- ponding to the minimal energy, which in the Kohn-Sham picture implies the minimisation
of EKS(Θ) with respect to the orbitals, thus the problem
E0≤ E0KS= inf n EKS(Θ) Θ ∈ H1(R3, C)Nelec ,∀i, j hψi|ψji1= δij. o . (4.99)
Both the energy functional (4.98) as well as the Kohn-Sham minimisation problem (4.99) are closely related to the HF problem (4.40). In fact the only difference is the substitution of the exchange energy term by the exchange-correlation functional. As such it should not be very surprising that the methods employed to solve (4.99) is very similar to HF as well. The conditions to obtain the stationary points of (4.99), the Euler-Lagrange equations, can be reformulated as
ˆ FKS Θ0ψ0i = εiψ0i and ψi0 ψ0 j = δij (4.100)
4.6. DENSITY-FUNCTIONAL THEORY 83
where Θ0 is the minimiser of (4.99) and
ˆ
FKS
Θ0 = ˆT + ˆV0+ ˆJΘ0+ Vxc (4.101)
is the Kohn-Sham operator. Its difference to the Fock operator (4.49) is again simply the
replacement of the exchange operator ˆKΘ0 by the exchange-correlation potential
Vxc(r), which is the derivative of the exchange-correlation energy Exc(ρ) with respect
to the density function ρ. Equation (4.100) as well as the minimisation problem (4.99) can now be discretised similar to the procedure outlined in section 4.4.1 on page 64 for Hartree-Fock, which leads to an iterative self-consistent field procedure, which is very similar to the Hartree-Fock SCF outlined in remark 4.18 on page 70. Algorithmically both for Kohn-Sham DFT as well as HF the same type of problem needs to be solved, such that all of the numerical procedures discussed in the next chapters for HF could be applied to Kohn-Sham DFT with only very few changes.
Even though the mathematical problem of the Kohn-Sham DFT ansatz is related to HF, one should mention that DFT in combination with modern exchange-correlation functionals [123–128] is much more exact than HF for common applications of quantum- chemical calculations. Since the cost is comparable to HF, it has thus become by far the most widely used method of electronic structure theory.
Chapter 5
Numerical approaches for
solving the Hartree-Fock
problem
I believe there is no philosophical high-road in science, with epistem- ological signposts. No, we are in a jungle and find our way by trial and error, building our road behind us as we proceed.
— Max Born (1882–1970)
This chapter is devoted to an in-depth discussion of numerical approaches for solving the HF problem both when it comes to the basis function type used for the discretisation and the algorithms for solving the discretised problem. We will discuss how different basis function types lead to numerical problems of vastly different structure and how therefore not every algorithmic ansatz works for every type of basis function.
In section 4.4.1 we noted that there are roughly three ways to view the discretised HF problem. One way would be to think of it as a minimisation of the energy with respect to the orbital coefficients, another as a minimisation with respect to the density matrix and yet a third as a non-linear eigenproblem, which needs to be solved self-consistently. Our discussion here will generally take the third viewpoint and only switch to the others when this aids our argument. Furthermore we will implicitly assume a real-valued UHF ansatz in this chapter. The adaption of the presented results to RHF or ROHF is usually straightforward1.
1 To go from UHF to RHF one just needs to consider both blocks of the relevant Fock, coefficient,
and density matrices to be equivalent. Going from UHF to ROHF only amounts to replacing the UHF Fock matrix by the appropriately constructed ROHF Fock matrix before performing the diagonalisation for getting the new coefficients.
86 CHAPTER 5. NUMERICAL APPROACHES FOR SOLVING HF
5.1
Overview of the self-consistent field procedure
In remark 4.18 of the previous chapter we suggested a simple procedure for iteratively
solving the HF equations. The idea was to start from an initial guess C(0)from the Stiefel
manifold C as defined in (4.66) and repetitively construct occupied coefficient matrices
C(1), C(2), . . . , C(n−1)
∈ C by solving the discretised HF equations (4.79) and considering
the Aufbau principle. Since the minimiser of the discretised HF problem (4.65) is unique2,
there is no need to diagonalise exactly FhC(n) C(n)†i in each iteration. Instead we
could well diagonalise an arbitrary matrix ˜F(n)for obtaining the new coefficients C(n). It
is important, however, to ensure that the final coefficients, say C0, satisfy the necessary
conditions for being a minimiser of EC, namely that C0 ∈ C and that the Pulay error
(4.80) vanishes. At least its norm should stay within a finite value. Notice, that for
the computation of the Pulay error in each case the unmodified matrix FhC(n) C(n)†i
needs to be employed in order for the resulting value to be meaningful. As discussed in remark 4.18 on page 70 even if both these conditions are satisfied, this is no guarantee,
however, that C(n) is a minimiser for (4.65), since both are only necessary but no
sufficient conditions. Ignoring this fact for a moment, this leads to the following general approach.
Remark 5.1(SCF procedure). Pick a convergence threshold εconv∈ R, a basis set
{ϕµ}µ∈Ibas ⊂ H
1(R3, R) and an initial guess C(0) ∈ C of occupied coefficients. From
this build an initial Fock matrix ˜F(0)= FhC(0) C(0)†i.
For n = 1, 2, 3, . . . • Diagonalise
˜
F(n−1)C(n)F = SC(n)F E(n)
under the condition
C(n)F † SC(n)F = INorb where E(n)= diagε(n)1 , ε (n) 2 , . . . , ε (n) Norb is the diagonal matrix of orbital eigenvalues.
• Construct the occupied matrix C(n) from the full matrix C(n)
F by the Aufbau
principle.
• Build the Fock matrix FhC(n) C(n)†i.
• Compute e(n)according to (4.80) e(n)= F C(n)C(n) † C(n)C(n) † S− S C(n)C(n) † F C(n)C(n) †
Check the necessary condition: If e(n)
frob ≤ εconv the procedure is considered
converged3 with final coefficients C
0≡ C(n).
2
This is only true in the discrete setting.
3