Solving CI for Correlated Ground States

Hello! Welcome back to our journey into many-body quantum theory.

In our last lesson, we established a systematic way to construct a basis for the N-electron wavefunction. We started with the Hartree-Fock (HF) ground state, $|{\Phi }_0⟩$, and then generated a hierarchy of excited Slater determinants—singles ($|{\Phi }_a^r⟩$), doubles ($|{\Phi }_{ab}^{rs}⟩$), and so on—by promoting electrons from occupied to virtual orbitals. The exact wavefunction within a given one-particle basis, $|\Psi ⟩$, can be written as a linear combination of all these determinants, an expansion known as Full Configuration Interaction (FCI).

Today, we will make this concrete. Our goal is to solve the Configuration Interaction (CI) problem for a minimal basis system to determine the correlated ground state. We will move from the abstract expansion $|\Psi ⟩={\sum }_I{c}_I|{\Phi }_I⟩$ to a practical calculation. We'll apply the variational principle to find the coefficients $c_I$ and the system's energy. This will lead us to set up and solve a matrix eigenvalue problem, providing our first tangible calculation of the electron correlation energy.

1. From Variational Principle to Matrix Eigenvalue Problem

The core idea of CI is to use the linear expansion of the wavefunction as a variational ansatz. Applying the variational principle to minimize the energy expectation value $E=\frac{⟨\Psi |\hat{H}|\Psi ⟩}{⟨\Psi |\Psi ⟩}$ with our ansatz $|\Psi ⟩={\sum }_I{c}_I|{\Phi }_I⟩$ leads to a set of secular equations.

In matrix form, this is written as:

where:

• $\mathbf{c}$ is the column vector of the coefficients $c_I$.
• $\mathbf{H}$ is the Hamiltonian matrix, with elements $H_{IJ}=⟨{\Phi }_I|\hat{H}|{\Phi }_J⟩$.
• $\mathbf{S}$ is the overlap matrix, with elements $S_{IJ}=⟨{\Phi }_I|{\Phi }_J⟩$.

A crucial simplification arises because the Slater determinants $|{\Phi }_I⟩$ are constructed from an orthonormal set of molecular spin-orbitals. As a result, the determinants themselves form an orthonormal basis: $S_{IJ}={\delta }_{IJ}$. The overlap matrix $\mathbf{S}$ becomes the identity matrix, and our problem reduces to the standard eigenvalue equation for the Hamiltonian matrix:

Solving this equation—that is, diagonalizing the CI matrix $\mathbf{H}$—yields a set of eigenvalues $E_k$ (the energies of the ground and excited states of the system) and their corresponding eigenvectors ${\mathbf{c}}_k$ (the coefficients defining the wavefunctions $|{\Psi }_k⟩$). The lowest eigenvalue, $E_0$, is the variationally optimal ground state energy.

2. A Concrete Case: The H₂ Molecule with a Minimal Basis

Let's apply this machinery to the simplest chemically-bonded system: the hydrogen molecule, H₂. We'll use a minimal basis set, which consists of one 1s atomic orbital on each hydrogen atom, ${\chi }_A$ and ${\chi }_B$.

A Hartree-Fock calculation on this system yields two molecular spatial orbitals:

• A bonding orbital: ${\sigma }_g\propto ({\chi }_A+{\chi }_B)$
• An anti-bonding orbital: ${\sigma }_u\propto ({\chi }_A-{\chi }_B)$

For the H₂ ground state, we have $N=2$ electrons.

1. The HF Reference Determinant ($|{\Phi }_0⟩$): The lowest energy configuration places both electrons in the bonding orbital with opposite spins. In our shorthand notation, this is $|{\sigma }_g^2⟩$. $$|{\Phi }_0⟩=|{\sigma }_g\alpha ,{\sigma }_g\beta ⟩$$
2. Excited Determinants: What other determinants can we form?
   • Single Excitations: We could promote one electron from ${\sigma }_g$ to ${\sigma }_u$, forming configurations like $|{\sigma }_g\alpha ,{\sigma }_u\beta ⟩$. However, the ground state $|{\Phi }_0⟩$ has a specific overall symmetry (gerade, or 'g'). The singly-excited states have a different symmetry (ungerade, or 'u'). Since the Hamiltonian $\hat{H}$ is totally symmetric, matrix elements between states of different symmetry are zero: $⟨{\Phi }_0|\hat{H}|{\Phi }_{\text{single}}⟩=0$. This is a consequence of Brillouin's theorem. These states don't mix with the ground state, so we can ignore them for our purpose of finding the correlated ground state.
   • Double Excitations: We can promote both electrons from ${\sigma }_g$ to ${\sigma }_u$. This creates the determinant $|{\Phi }_D⟩=|{\sigma }_u^2⟩$. This state has the same overall 'g' symmetry as the ground state and can mix with it.
   $$|{\Phi }_D⟩=|{\sigma }_u\alpha ,{\sigma }_u\beta ⟩$$
   • Higher excitations are not possible since we only have two electrons.

Therefore, for this minimal system, the Full CI wavefunction is a linear combination of just two determinants:

Our task is to solve the 2x2 eigenvalue problem for this two-state system.

3. Constructing and Solving the CI Matrix

As explained in the video, we need to build and diagonalize the following 2x2 matrix:

Let's break down the matrix elements:

• $H_{00}$: This is the expectation value of the Hamiltonian for the Hartree-Fock reference state. By definition, this is the Hartree-Fock energy, $E_{HF}$.
• $H_{DD}$: This is the energy of the doubly excited configuration, $⟨{\sigma }_u^2|\hat{H}|{\sigma }_u^2⟩$. This is some higher energy value.
• $H_{0D}=H_{D0}^{\ast }$: This is the coupling term or off-diagonal matrix element. It represents the interaction between the HF ground state and the doubly excited configuration. According to the Slater-Condon rules for evaluating matrix elements between determinants that differ by two spin-orbitals, this simplifies to a single two-electron integral: $$H_{0D}=⟨{\Phi }_0|\hat{H}|{\Phi }_D⟩=⟨{\sigma }_g{\sigma }_g|r_{12}^{-1}|{\sigma }_u{\sigma }_u⟩\equiv K_{gu}$$ This is an exchange-like integral between the bonding and anti-bonding orbitals. It is this non-zero coupling that allows the configurations to "interact" and mix.

The eigenvalue problem is then to solve the secular determinant:

This is a quadratic equation for the energy $E$. The two solutions (roots) for the energy are:

To see how CI improves upon Hartree-Fock, focus on the lowest energy root, $E_{-}$. As long as the coupling $K_{gu}$ is non-zero, the term under the square root is strictly greater than ${\left(\frac{E_{HF}-H_{DD}}{2}\right)}^2$. This means we subtract a quantity larger than $\frac{H_{DD}-E_{HF}}{2}$ from the average energy $\frac{E_{HF}+H_{DD}}{2}$. The result is that the new ground state energy $E_{-}$ is always lower than $E_{HF}$.

We have successfully captured some of the electron correlation!

The correlation energy ($E_c$) is formally defined as the difference between the exact energy and the Hartree-Fock energy. In our case, the Full CI gives the exact energy within this basis, so:

By plugging $E_{-}$ into the eigenvalue equation, we can also solve for the eigenvector $\mathbf{c}=(c_0,c_D)$, which gives us the improved, correlated ground state wavefunction: $|{\Psi }_{g.s.}⟩=c_0|{\Phi }_0⟩+c_D|{\Phi }_D⟩$.

4. A Critical Limitation: The Size-Extensivity Problem

The H₂ example is a "Full CI" because we included all possible determinants. For any larger system, Full CI is computationally impossible due to the combinatorial explosion we discussed last time. The practical approach is to truncate the CI expansion, for instance, by including only single and double excitations (CISD).

However, this truncation introduces a serious theoretical flaw known as the size-extensivity problem. A method is size-extensive if the energy of a system of N non-interacting particles is equal to N times the energy of a single particle. Truncated CI is not size-extensive.

This failure to scale correctly with system size is a major drawback of truncated CI methods and is a primary motivation for the development of alternative approaches like Møller-Plesset perturbation theory and, most importantly, Coupled Cluster theory, which we will touch upon in a later module.

Conclusion

In this lesson, we bridged the gap between the formal theory of the many-body wavefunction and a concrete, solvable problem. We saw how the variational principle transforms the CI problem into one of matrix diagonalization.

Key Takeaways:

• Configuration Interaction is a variational method that finds the optimal linear combination of Slater determinants.
• The method involves solving the matrix eigenvalue equation $\mathbf{Hc}=E\mathbf{c}$, where the basis functions are the determinants $|{\Phi }_I⟩$.
• For the H₂ molecule, a 2x2 CI problem demonstrates how mixing the Hartree-Fock ground state with a doubly excited state lowers the total energy, thus capturing correlation energy.
• The off-diagonal matrix elements $⟨{\Phi }_I|\hat{H}|{\Phi }_J⟩$ are crucial as they govern the mixing between configurations.
• While Full CI is exact (for a given basis), practical truncated CI methods like CISD are not size-extensive, which is a significant theoretical limitation.

Preview of the Next Lesson:

We have now concluded our deep dive into the Hartree-Fock and Configuration Interaction methods, which form the cornerstone of wave-function based quantum chemistry. We will now pivot to an entirely different, yet equally powerful, formalism for tackling the many-body problem, one that is particularly prevalent in condensed matter physics. In the next module, we will be introduced to Many-Body Perturbation Theory, starting with the definition and physical interpretation of single-particle Green's functions. This will set the stage for understanding Feynman diagrams and the concept of quasiparticles.