Finite tensor contractions and their variational energy and gradient

Let $n,r\ge 1$ be integers, $C\in {\mathbb{R}}^{n\times r}$, $D\in {\mathbb{R}}^{r\times r}$, $H\in {\mathbb{R}}^{n\times n}$, and $G\in {\mathbb{R}}^{n\times n\times n\times n}$. Define their transformed tensors by $H_{pq}^{\mathrm{\prime }}={\sum }_{a,b=1}^n{C}_{ap}{H}_{ab}{C}_{bq},G_{pqrs}^{\mathrm{\prime }}={\sum }_{a,b,c,d=1}^n{C}_{ap}{C}_{bq}{C}_{cr}{C}_{ds}{G}_{abcd},$ for $1\le p,q,r,s\le r$. Define the finite variational energy functional $E(C;D,H,G)={\sum }_{p,q=1}^r{D}_{pq}{H}_{pq}^{\mathrm{\prime }}+\frac{1}{2}{\sum }_{p,q,r,s=1}^r{D}_{pq}{D}_{rs}{G}_{pqrs}^{\mathrm{\prime }}\mathrm{.}$ Its gradient ${\mathrm{\nabla }}_{C}E$ is the ordinary Euclidean gradient with respect to the $nr$ real entries of $C$. The basis dimension is $n$, the contracted dimension is $r$, and for any finite matrix or tensor $A$, $\mathrm{\parallel }A{\mathrm{\parallel }}_F=(\sum \mathrm{\mid }{A}_{i_1\dots i_k}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, with the sum over all its entries. This is a finite algebraic functional; it assumes no Hartree–Fock, self-consistency, or minimizing-state postulate.

For perturbed integral tensors $\tilde{H},\tilde{G}$, write $\tilde{E}$ for the same functional with those tensors and the same fixed $C,D$. The word “variational” here names the finite coefficient functional only; no claim about a stationary point or physical energy is included.