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Hermite recurrence formulas for Gaussian nuclear-attraction and two-electron Coulomb integrals

Statement

For $p > 0$ , $P, C \in R^{3}$ , and $η = p q / (p + q)$ with $p, q > 0$ , the Hermite Coulomb auxiliaries of Hermite Gaussian and Boys Coulomb auxiliary functions satisfy $R_{000}^{n} (p, P - C) = (- 2 p)^{n} F_{n} (p ∥ P - C ∥^{2}),$ $R_{t + 1, u, v}^{n} = (P_{1} - C_{1}) R_{t u v}^{n + 1} + t R_{t - 1, u, v}^{n + 1},$ with the corresponding coordinate formulas for increasing $u$ or $v$ and with negative-index terms zero. The one-center and two-center base Coulomb integrals are $\int_{R^{3}} \frac{e^{- p ∥ x - P ∥^{2}}}{∥ x - C ∥} d x = \frac{2 π}{p} F_{0} (p ∥ P - C ∥^{2}),$ $\int ⁣ ⁣ \int \frac{e^{- p ∥ x - P ∥^{2}} e^{- q ∥ y - Q ∥^{2}}}{∥ x - y ∥} d y d x = \frac{2 π^{5 / 2}}{p q \sqrt{p + q}} F_{0} (η ∥ P - Q ∥^{2}) .$ For a primitive pair expansion $g_{a} g_{b} = \sum_{t u v} E_{a b}^{t u v} Λ_{t u v}$ , the nuclear attraction integral is $V_{C} (g_{a}, g_{b}) = \frac{2 π}{p} \sum_{t u v} E_{a b}^{t u v} R_{t u v}^{0} (p, P - C) .$ For two primitive pairs, with coefficients $E_{a b}$ and $E_{c d}$ and centers $P, Q$ , the Coulomb integral is $G (g_{a}, g_{b}; g_{c}, g_{d}) = \frac{2 π^{5 / 2}}{p q \sqrt{p + q}} \sum_{t u v} \sum_{τ ν ϕ} \overline{E_{a b}^{t u v}} E_{c d}^{τ ν ϕ} (- 1)^{τ + ν + ϕ} R_{t + τ, u + ν, v + ϕ}^{0} (η, P - Q) .$ All sums are finite; the largest Boys order is bounded by the total polynomial degree in the participating primitives.

Facts & Assumptions

Given: Primitive Gaussians and Lebesgue integral conventions from Cartesian Gaussian primitives, contractions, and integral conventions, Hermite and $R^{n}$ definitions from Hermite Gaussian and Boys Coulomb auxiliary functions, and finite pair expansions from Finite Hermite expansion and coefficient recurrences for a Cartesian Gaussian pair.

[F1]

Gaussian product theorem and finite polynomial shifts: $e^{- a ∥ x - A ∥^{2}} e^{- b ∥ x - B ∥^{2}} = e^{- a b ∥ A - B ∥^{2} / (a + b)} e^{- (a + b) ∥ x - P ∥^{2}}$ with $P = (a A + b B) / (a + b)$ .

[F2]

Finite Hermite expansion and coefficient recurrences for a Cartesian Gaussian pair: each Cartesian primitive pair has a finite Hermite-Gaussian expansion with the stated coefficients.

[F3]

Hermite Gaussian and Boys Coulomb auxiliary functions: $Λ_{t u v} = \partial_{P}^{t u v} e^{- p ∥ x - P ∥^{2}}$ and $R_{t u v}^{n} = (- 2 p)^{n} \partial_{P}^{t u v} F_{n} (p ∥ P - C ∥^{2})$ .

[F4]

Boys integral, power series, recurrence, and incomplete-gamma representation: $F_{n}^{'} (T) = - F_{n + 1} (T)$ and $F_{n} (T) = \int_{0}^{1} t^{2 n} e^{- T t^{2}} d t$ .

[F5]

The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ and Cartesian Gaussian moments and exponent-parameter derivatives: $\int_{R} e^{- x^{2}} d x = \sqrt{π}$ and $\int_{R^{3}} e^{- p ∥ x ∥^{2}} d x = (π / p)^{3 / 2}$ .

[F6]

Absolute convergence and local parameter domination for Gaussian basis integrals: the Coulomb integrals and all finite center derivatives are absolutely integrable with a common local parameter majorant.

[F7]

Change of variable in an improper integral: a monotone differentiable surjection preserves convergent improper integrals when its derivative and the target integrand are locally Riemann integrable and the proper substitution hypotheses hold on every compact truncation, with singular ends treated separately.

[L8]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product and Fubini's theorem for L^1 functions on a sigma-finite product permit interchange of nonnegative and absolutely integrable iterated integrals, respectively.

[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

1.1givenF1F5F7L8algebra

For $r > 0$ , the Gaussian integral and substitution $u = s r$ give $r^{- 1} = (2 / \sqrt{π}) \int_{0}^{\infty} e^{- s^{2} r^{2}} d s$ . Insert this representation into the one-center integral; Tonelli applies to the nonnegative integrand. Completing the square in $x$ using [F1] gives a three-dimensional Gaussian integral $(π / (p + s^{2}))^{3 / 2}$ times $e^{- p s^{2} ∥ P - C ∥^{2} / (p + s^{2})}$ . The substitution $t = s / \sqrt{p + s^{2}}$ maps $[0, \infty)$ monotonically onto $[0, 1)$ and simplifies the remaining factor to $2 π / p \int_{0}^{1} e^{- p ∥ P - C ∥^{2} t^{2}} d t$ , which is the stated base formula. Both substitutions are smooth monotone surjections with continuous derivatives and integrands on their compact truncations; the transformed Boys integrand is bounded at $t = 1$ , so [F7] applies at the improper ends.

1.2givenF3F4L9algebrainduction

From [F4] and the chain rule, $\partial_{P_{1}} F_{n} (p ∥ P - C ∥^{2}) = - 2 p (P_{1} - C_{1}) F_{n + 1} (p ∥ P - C ∥^{2})$ . Applying $\partial_{P_{1}}^{t} \partial_{P_{2}}^{u} \partial_{P_{3}}^{v}$ and the product rule to $(P_{1} - C_{1}) F_{n + 1}$ produces one term $(P_{1} - C_{1}) R_{t u v}^{n + 1}$ and $t$ terms $R_{t - 1, u, v}^{n + 1}$ . Multiplying by the normalization $(- 2 p)^{n}$ yields the displayed recurrence; the same argument applies to the other coordinates.

2.1givenF1F5F6L8step 1.1algebra

For the two-center base, set $z = x - y$ and integrate first in $x$ . Completing the square with [F1] gives $\int_{R^{3}} e^{- p ∥ x - P ∥^{2}} e^{- q ∥ x - z - Q ∥^{2}} d x = (π / (p + q))^{3 / 2} e^{- η ∥ z - (P - Q) ∥^{2}} .$ Absolute convergence [F6] justifies the change of variables and order of integration. Applying step 1.1 with exponent $η$ and center difference $P - Q$ , then simplifying $(π / (p + q))^{3 / 2} (2 π / η)$ , gives $2 π^{5 / 2} / (p q \sqrt{p + q})$ times the stated Boys value.

3.1givenF3F6step 1.1step 2.1algebraL9

By [F6], differentiating either base integral with respect to its Gaussian centers is valid for every finite derivative order. Since $Λ_{t u v}$ is exactly the corresponding center derivative by [F3], the one-center Hermite Coulomb integral is $(2 π / p) \partial_{P}^{t u v} F_{0} = (2 π / p) R_{t u v}^{0}$ . In the two-center case, derivatives in $Q$ act on $P - Q$ with a minus sign for each coordinate derivative; hence the Hermite integral is the base constant times $(- 1)^{τ + ν + ϕ} \partial_{P - Q}^{t + τ, u + ν, v + ϕ} F_{0}$ .

4.1givenF1F2step 3.1algebra∎

Substitute the finite expansions from [F2] into the one- and two-center integrals. Linearity makes the sums finite. The one-center Hermite formula from step 3.1 gives the stated nuclear formula, and the two-center formula from step 3.1 gives the displayed signed double sum. Complex contractions use conjugate-linear coefficients in the first pair exactly as in the Coulomb form specified in Given; the primitive formulas are real.

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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)