As mentioned before, approaches we desire can be somewhat application-specific.
We have enough tools to apply to a wide variety of problems, though they may become too costly for large systems. Once we have a method that is targeted to solve a specific class of problems of interest, it is often necessary to have additional tools which could determine whether a given problem can be tackled by the method.
Development of such tools that help one choose the right approach is therefore highly desirable.
Distinguishing Weak and Strong Correlation
In Section 1.1, we introduced weak and strong correlation between electrons.
Strong correlation is typically difficult to deal with and the applicability of available tools are highly limited in system size. On the other hand, we have many available tools for weak correlation such as PT, which can potentially be applied to systems of thousands of atoms with additional tricks such as locality of weak correlation.
Most approaches to strong correlation problems consist of two steps. First, one builds an “active space” problem, which is to form an effective Hamiltonian for all strongly correlated electrons, and solves this Hamiltonian exactly (or often approx-imately with high accuracy). Second, one adds weak correlation outside this active space via a CI expansion [68–72], a CC expansion [73–111], or PT [112–119]. The first step of solving an active space problem is normally limited in size as most available tools scale exponentially with the number of electrons. The second step of adding missing weak correlation formally scales polynomially but is usually very complicated in terms of implementation. As such, the resulting model chemistry lacks derivation and implementation of properties beyond energies. If a given system
is only weakly correlated, the situation is much simpler. One can apply a low-order perturbation approach based on a mean-field starting point, Hartree-Fock (HF), and obtain quantitative answers for energies and other properties. Various properties are readily available for simple PT such as second-order Møller-Plesset (MP2) theory.
This is due to the fact that our starting point for PT here is a single determinant.
In the case of the second step for strong correlation we started from a more com-plicated active space wavefunction, which poses challenges in implementation and extension of such PT methods. Given this situation, it is preferred to distinguish weak and strong correlation and apply the two-step-strategy of strong correlation to only strongly correlated systems. For weakly correlated systems, such an approach is mostly wasteful and we should attempt to apply a simple approach like MP2.
In the modern era of quantum chemistry, distinguishing these two were normally done by symmetry breaking at the HF level. The meaning of this is most apparent in the HF variational energy expression for N-electron systems:
EHF= inf
where |Φi is a single Slater determinant made of a set of occupied orbitals {|φii}, the one-electron operator ˆh is
ˆh =−1
2∇2−X
A
ZA
|r − RA|, (1.7)
the antisymmetrized two-electron integral, hij||iji, reads
hij||iji ≡ hij|iji − hij|jii (1.8) with the Coulomb term,
and Λ represents the Lagrange multipliers to enforce orthonormality between or-bitals. Eq. (1.6) demonstrates the variational nature of HF. Namely, the HF method produces a variationally optimized Slater determinant that gives an infimum of the energy expression in Eq. (1.6). In practice, one might impose “symmetry con-straints” on the form of|Φi to obtain a determinant with the same symmetry as the exact ground state of ˆHel. The exact ground state has multiple symmetry properties because ˆHel commutes with other operators. The symmetry of exact ground state includes ˆSz-symmetry, spin- ( ˆS2-) symmetry, complex symmetry, and time-reversal symmetry. The symmetry constraints can be understood in terms of working quanti-ties, C (i.e., molecular orbital coefficients). C is used to define each molecular orbital (MO) as a linear combination of atomic orbitals (LCAO), that is,
φi(r) =X
µ
ηµ(r)Cµi (1.11)
where ηµ(r) denotes an atomic orbital (AO) evaluated at a grid point r. The matrix C is a transformation matrix that connects MOs with AOs. A most general form of the i-th column (i.e., the i-th MO) in C is
cαi cβi
(1.12) where α and β denote the spin-component such that
φi(r) =X
µ
((cαi)µηµ(r) + (cβi)µηµ(r)). (1.13)
A one-body reduced density matrix (1RDM) P is then composed of a total of four spin-blocks,
P = CoccC†occ =Pαα Pαβ Pβα Pββ
(1.14) where Cocc is the MO coefficient matrix for the occupied orbitals. Possible symmetry constraints on|Φi can be understood as restrictions on P. The necessary constraints for representing non-interacting N-electron systems are (1) tr(P ) = Nel and (2) P2 = P (assuming AO basis sets are orthogonal). Depending on the type of HF, one may impose additional constraints on P. In Table 1.1, we present possible HF types along with their restrictions on P. Only RHF possesses all symmetries that the exact ground state has for a given Hamiltonian ˆHel. The common wisdom in modern quantum chemistry is that strong correlation exists when RHF is unstable to other HF types. This common wisdom has been challenged by multiple examples [120–
Type Constraints
RHF Pαα = Pββ, Pαβ = Pβα= 0, Pµναα ∈ R cRHF Pαα = Pββ, Pαβ = Pβα = 0
UHF Pαβ = Pβα = 0, Pµναα ∈ R, Pµνββ ∈ R
cUHF Pαβ = Pβα = 0
GHF Pµν ∈ R
cGHF None
Table 1.1: Types of HF approaches and their constraints on 1RDM. Note that we assume that orbitals are maximally collinear along the z-axis. For more general classifications without assuming the z-axis being a special axis, see Ch. 3.
127] that often call this “artificial” symmetry breaking. Artificial symmetry breaking occurs due to the lack of weak correlation, not due to the lack of strong correlation.
On the contrary, essential symmetry breaking is due to the lack of strong correlation at the HF level and this implies that we need an approach that goes beyond simple PT methods. This artificial symmetry breaking cannot be distinguished from “essential”
symmetry breaking solely based on the HF theory. We address this issue based on a correlated orbital theory where orbitals are optimized in the presence of weak correlation (in this case MP2). The resulting method may be useful for treating molecular problems without artifacts from artificial symmetry breaking using HF orbitals, as well as signaling the onset of genuinely strong correlations through the presence of essential symmetry-breaking in its reference orbitals
Exact, Polynomial Description to Strong Spin Correlation
We developed an approach that can describe strong spin correlation (SSC) ex-actly without invoking an exponential cost wall. SSC is an important class of strong correlation. When there are many spatially separated open-shell electrons, there is only a small energy cost for spin-flips while charge-transfers exhibit a large energy cost. This is the defining property of SSC [128]. There are multiple familiar exam-ples that show emergent SSC between electrons. The most widely known example is multiple bond dissociations in molecules. For instance, dissociating the C-C double bond in an ethene molecule (H2C−−CH2) leads to a total of four open-shell electrons that participate in SSC of this molecule. More sophisticated examples include metal-loenzymes [129–131] such as the P-center of nitrogenase which involves the so-called FeMoCo moiety [132–135]. Unlike bond dissociations, metalloenzymes exhibit many open-shell electrons even at their equilibrium geometries. This is largely due to the fact that d- or f-electrons tend to be localized in space. Some metalloenzymes may
involve hundreds of open-shell electrons that participate in SSC and thus there is a clear need for an economical method which can handle SSC efficiently.
The common wisdom in modern quantum chemistry is that SSC can be described exactly if one includes all possible spin-couplings for a given number of electrons, N , and a spin-quantum number S. This is the essence of the spin-coupled valence bond (SCVB) or generalized valence bond (GVB) approach. It is undoubtedly the right approach for SSC but the cost for including all possible spin-couplings in SCVB scales exponentially with the number of electrons. The number of independent spin-coupling vectors is given by
f (N, S) = (2S + 1)N !
(12N + S + 1)!(12N − S)!, (1.15) assuming the ˆSz quantum number M = S [136]. Based on Sterling’s approximation, this combinatorial scaling of the spin-coupling dimension can be shown to approach an asymptotic exponential scaling for large N . Similar to other exponential scaling approaches, due to the steep computational scaling, the applicability of SCVB has been limited to systems with a small number of electrons [137–139].
Our approach to this seemingly exponentially difficult problem is the coupled-cluster valence bond (CCVB) approach [140]. We cast the SCVB wavefunction, which contains an exponentially many spin-couplings, into a compact coupled-cluster expansion with only a quadratic number of parameters, {tKL}. The resulting wave-function from CCVB is size-consistent, spin-pure (i.e., it is a spin-eigenwave-function) and exact for SSC (i.e., exact for multiple bond breaking). Furthermore, the overall cost of the CCVB energy evaluation along with the wavefunction optimization scales only cubically with system size. It turns out that CCVB can describe SSC exactly as long as the UHF energy (see Table 1.1) is exact within the valence active space (where chemical bonds form and break). The UHF energy is exact in this limit because states with different S are all degenerate and all the interacting electrons are high-spin (i.e., no correlation). In such cases, CCVB is exact and maintains the spin-purity.
Some drawbacks and challenges in CCVB were addressed in this dissertation.
Those challenges are as follows:
1. The CCVB energy is not invariant under unitary transformations within the occupied-occupied and the virtual-virtual blocks. This makes CCVB highly non-trivial to use even for an expert. If there is a way to improve, it is highly desirable.
2. The CCVB active space is inherently limited to a pairing active space (N e, N o) where we have the same number of electrons and the same number of orbitals
for CCVB to solve. This limits the applicability of CCVB to problems like bond dissociations where such a pairing active space is naturally well-suited.
3. CCVB does not include any ionic excitations between electron pairs. These ionic excitations do not participate in SSC, but they play a crucial role in describing overall electronic structure for systems with mixed SSC and weak correlation.
4. Some problems with SSC are not treatable with CCVB. These examples are those that UHF fails and only cGHF (see Table 1.1) is well-suited among HF approaches.
The drawbacks 1–3 have been addressed by extending CCVB to CCVB with singles and doubles (CCVB-SD) [141, 142] and the drawback 4 have been addressed by extending CCVB further to incorporate missing spin-couplings[143, 144].