What is the X in Almlöf and Taylor's "Unified treatment of energy derivatives?"

Question

I have been studying the possible methods for basis set optimizations. One notable paper is "Energy-optimized GTO basis sets for LCAO Calculations. A Gradient Approach" by Knut Faegri Jr. and Jan Almlof (link). In it, they define and use an approach which involves a transformation $T$ decomposed as: $$T = \exp(X) \exp(Y) \tag{1}$$ where $\exp(X)$ is a unitary transformation and $X$ is an anti-Hermitian matrix. However, the paper does not go into detail about how $X$ can be obtained and used in an implementation of the algorithm. Additionally, a paper which they cite, "Molecular properties from perturbation theory: A Unified treatment of energy derivatives" by Jan Almlöf and Peter R. Taylor (link) provides little further detail, by defining $U = \exp(X)$, but not providing details how $X$ can be calculated here. Does anyone have a good reference for this method, or is able to provide more background on the definitions which are missing?

Answer 1

Question

"Jan Almlöf and Peter R. Taylor (link) provides little further detail, by defining U=exp(X), but not providing details how X can be calculated here."

Answer

$X$ is literally defined by:

$$U = \exp(X)\tag{1},$$

so if you want $X$, you can calculate it by computing a natural logarithm of $U$. Unlike $X$ which is defined based on $U$, the operator $U$ has much more of a "meaning". This is all explained in the same paragraph from which you found Eq. 1 (which is Eq. 42 in the paper)! If $U$ want an explicit expression for $U$ in terms of "known" quantities, then I've provided it in Eq. (7) below.

More details

A Hamiltonian $\hat{H}_0$ is perturbed by $\hat{V}(\lambda)$:

$$ \tag{2} \hat{H} = \hat{H}_0 + \hat{V}(\lambda), $$

which causes the unperturbed molecular orbital coefficients $C_0$ to get perturbed into $C(\lambda)$ by the linear transformation $T(\lambda)$ as stated in Eq. 40 of the paper (Eq. 40 also writes $C_0$ as having an explicit dependence on $\lambda$ but I don't yet see that being necessary):

$$ \tag{3} C(\lambda) = C_0 T(\lambda), $$

in which (Eq. 41 of the paper): $$\tag{4} T(\lambda) = V(\lambda)U(\lambda), $$

with $V(\lambda)$ being explicitly given in terms of the AO overlap matrix $S$ in Eq. 43:

$$\tag{5} V(\lambda) = \left(C_0^\dagger S(\lambda) C_0\right)^{-1/2}. $$

This means that if you have MO coefficients stored in $C_0$ and you perturb the Hamiltonian by $\hat{V}(\lambda)$, the new MO coefficients can be stored in:

$$C(\lambda) = C_0\left(C_0^\dagger S(\lambda) C_0\right)^{-1/2}U (\lambda),\tag{6}$$

or if you have the MO coefficients before and after the perturbation, then you can obtain $X$ by:

$$ \begin{align} U(\lambda) &= \left( \left(C_0^\dagger S(\lambda) C_0\right)^{-1/2}\right)^{-1} C_0^{-1} C(\lambda), \tag{7}\\ \exp\left(X(\lambda)\right) &\equiv \left( \left(C_0^\dagger S(\lambda) C_0\right)^{-1/2}\right)^{-1} C_0^{-1} C(\lambda), \tag{8}\\ X(\lambda) &= \ln \left(\left( \left(C_0^\dagger S(\lambda) C_0\right)^{-1/2}\right)^{-1} C_0^{-1} C(\lambda) \right),\tag{9}\\ X &= \ln \left(\left( \left(C_0^\dagger S C_0\right)^{-1/2}\right)^{-1} C_0^{-1} C \right).\tag{10} \end{align} $$

Two more minor notes:

• Eq. 55 of the paper and the paragraph directly before Eq. 53, tell us that in the case of an MCSCF wavefunction, of which HF wavefunctions (which are the only wavefunctions for which I've seen derivatives with respect to exponents done analytically in the papers up to the year of this Almlöf and Taylor paper) can be considered a "special case", there is no contribution to $\frac{\textrm{d}E}{\textrm{d}\lambda}$ from $\frac{\textrm{d}X}{\textrm{d}\lambda}$ so setting $\frac{\textrm{d}E}{\textrm{d}\lambda} = 0$ can be done by Eq. 56.
• The paper says "Most previous treatments (see, e.g., Ref. 9) have handled the MO orthonormality problem using Lagrange multipliers. This approach -- which must, of course, ultimately yield the same formulas as the present scheme -- is discussed briefly in Sec. V", so if you can get access to Ref. 9 which is "P. Pulay, in Applications of molecular Electronic Structure Theory, H. F. Schaefer 111, Ed. (Plenum, New York, 1977)", then you might be able to see all of this from another perspective.