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Finite Hermite expansion and coefficient recurrences for a Cartesian Gaussian pair

Statement

In one dimension, let $g_{i} (x_{A}, a) = x_{A}^{i} e^{- a x_{A}^{2}}$ and $g_{j} (x_{B}, b) = x_{B}^{j} e^{- b x_{B}^{2}}$ , with $x_{A} = x - A$ , $x_{B} = x - B$ , $p = a + b$ , $P = (a A + b B) / p$ , and $K = e^{- a b (A - B)^{2} / p}$ . There are unique coefficients $E_{i j}^{t}$ such that $g_{i} (x_{A}, a) g_{j} (x_{B}, b) = \sum_{t = 0}^{i + j} E_{i j}^{t} Λ_{t} (x; p, P) .$ With $E_{00}^{0} = K$ and $E_{i j}^{t} = 0$ outside $0 \leq t \leq i + j$ , they satisfy $E_{i + 1, j}^{t} = \frac{E_{i j}^{t - 1}}{2 p} + (P - A) E_{i j}^{t} + (t + 1) E_{i j}^{t + 1},$ $E_{i, j + 1}^{t} = \frac{E_{i j}^{t - 1}}{2 p} + (P - B) E_{i j}^{t} + (t + 1) E_{i j}^{t + 1} .$ In three dimensions the pair expansion is the product of the three one-dimensional expansions, so it is a finite sum of $Λ_{t u v}$ with $0 \leq t \leq α_{1} + β_{1}$ , $0 \leq u \leq α_{2} + β_{2}$ , and $0 \leq v \leq α_{3} + β_{3}$ .

Facts & Assumptions

Given: Primitive Gaussian factors and indices as above, the product identity of Gaussian product theorem and finite polynomial shifts, and $Λ_{t}$ from Hermite Gaussian and Boys Coulomb auxiliary functions.

[F1]

Gaussian product theorem and finite polynomial shifts: the product exponential equals $K e^{- p (x - P)^{2}}$ and the remaining polynomial has degree at most $i + j$ .

[F2]

Hermite Gaussian and Boys Coulomb auxiliary functions: $Λ_{t} = \partial_{P}^{t} e^{- p (x - P)^{2}}$ .

[L9]

The exponential function is smooth and $(\exp)^{'} = \exp$ and Sum, product, quotient, and chain rules from difference quotients: the exponential is smooth with derivative itself; finite sums, products and polynomial compositions obey the product, power and chain rules. Applying these rules one coordinate or parameter at a time justifies every displayed finite Gaussian derivative.

Proof

1.1givenF2algebrainductionL9

Differentiating [F2] once gives $Λ_{t + 1} = 2 p \partial_{P}^{t} ((x - P) e^{- p (x - P)^{2}})$ . The product rule applied $t$ times gives $\partial_{P}^{t} ((x - P) h) = (x - P) \partial_{P}^{t} h - t \partial_{P}^{t - 1} h$ , hence $(x - P) Λ_{t} = \frac{Λ_{t + 1}}{2 p} + t Λ_{t - 1},$ with the last term zero for $t = 0$ .

2.1givenF1F2step 1.1algebrainduction

The polynomial factors $x_{A}^{i} x_{B}^{j}$ have degree at most $i + j$ . By [F1] they multiply a common Gaussian. The polynomial part of $Λ_{t}$ has degree $t$ and leading coefficient $(2 p)^{t}$ , as follows inductively from $Λ_{t + 1} = 2 p (x - P) Λ_{t} - 2 p t Λ_{t - 1}$ . Thus these polynomials form a triangular basis of the polynomials of degree at most $i + j$ , which gives existence and uniqueness of the finite coefficients $E_{i j}^{t}$ .

3.1step 1.1step 2.1algebra

Multiplying the expansion for $g_{i} g_{j}$ by $x_{A} = (x - P) + (P - A)$ gives the expansion for $g_{i + 1} g_{j}$ . Substitute the identity in step 1.1 and collect the coefficient of each $Λ_{t}$ ; it is $E_{i j}^{t - 1} / (2 p) + (P - A) E_{i j}^{t} + (t + 1) E_{i j}^{t + 1}$ . Replacing $x_{A}$ by $x_{B} = (x - P) + (P - B)$ proves the second recurrence. The zero-outside-range convention handles both endpoints.

4.1givenF1step 2.1algebra∎

In three dimensions the primitive product factors into the product of its three one-dimensional coordinate factors. Multiplying their finite expansions gives the stated rectangular index range and a finite expansion in $Λ_{t u v}$ .

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Sources

Trygve Helgaker, Molecular Integral Evaluation, 10th Sostrup Summer School lecture notes (2008) (standard reference, not scraped)
L. E. McMurchie and E. R. Davidson, One and Two Electron Integrals over Cartesian Gaussian Functions, UCRL-13791 (1976) (standard reference, not scraped)