Cartesian Gaussian primitives, contractions, and integral conventions

Write ${\mathbb{N}}_0=\{0,1,2,\dots \}$ and $\mathrm{\parallel }x{\mathrm{\parallel }}_2=(x_1^2+x_2^2+x_3^2{)}^{1\mathrm{/}2}$ for $x\in {\mathbb{R}}^3$. For $\alpha =({\alpha }_1,{\alpha }_2,{\alpha }_3)\in {\mathbb{N}}_0^3$, exponent $a>0$, and center $A\in {\mathbb{R}}^3$, a primitive Cartesian Gaussian is $g_{\alpha ,a,A}(x)={\prod }_{j=1}^3(x_j-A_j{)}^{{\alpha }_j}\exp (-a\mathrm{\parallel }x-A{\mathrm{\parallel }}_2^2),x\in {\mathbb{R}}^3\mathrm{.}$ A finite Gaussian contraction is a finite sum $\varphi ={\sum }_{k=1}^q{c}_k{g}_{{\alpha }_k,a_k,A_k}$ with $q\in {\mathbb{N}}_0$, $c_k\in \mathbb{C}$, $a_k>0$, and $A_k\in {\mathbb{R}}^3$. The zero-term sum is the zero function. Bras are conjugated.

For a pair of contractions put $f_{\varphi ,\psi }(x)=\overline{\varphi (x)}\psi (x)$. For pair densities $f,g$, define the Coulomb form $\mathcal{C}(f,g)={\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{R}}^3}\frac{\overline{f(x)}g(y)}{\mathrm{\parallel }x-y{\mathrm{\parallel }}_2}dydx\mathrm{.}$ For finite contractions $\varphi ,\psi ,\rho ,\sigma$, define the overlap, kinetic, point-nucleus attraction, and two-electron Coulomb expressions by $S(\varphi ,\psi )={\int }_{{\mathbb{R}}^3}\overline{\varphi (x)}\psi (x)dx,T(\varphi ,\psi )=-\frac{1}{2}{\int }_{{\mathbb{R}}^3}\overline{\varphi (x)}\mathrm{\Delta }\psi (x)dx,$ $V_C(\varphi ,\psi )={\int }_{{\mathbb{R}}^3}\frac{\overline{\varphi (x)}\psi (x)}{\mathrm{\parallel }x-C{\mathrm{\parallel }}_2}dx,$ $G(\varphi ,\psi ;\rho ,\sigma )=\mathcal{C}(f_{\varphi ,\psi },f_{\rho ,\sigma })\mathrm{.}$ The primitives and their finite contractions are smooth: The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$ supplies smoothness of exp, and repeated coordinatewise applications of Sum, product, quotient, and chain rules from difference quotients supply all polynomial-product and composition derivatives. Thus the following Laplacian is well defined.

Here $\mathrm{\Delta }\psi ={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}^2\psi$ is the Cartesian Laplacian. All displayed integrals mean Lebesgue integrals in the sense of Integrable real and complex functions, and their integrals. A finite nuclear attraction is any finite real linear combination of the $V_C$ terms, with its centers and weights specified. This definition alone asserts neither convergence nor a physical approximation postulate.

The exponents, centers, polynomial multi-indices, coefficients, and nuclear data are explicit finite input data. A contraction coefficient may be complex; later real variational energy bounds state their separate real-matrix hypotheses.