Free-particle Green's function matrix elements over spherical Gaussian and plane-wave-modulated Gaussian basis functions

Dibyendu Mahato; Wojciech Skomorowski

arxiv: 2605.18564 · v1 · pith:FSQQGBAMnew · submitted 2026-05-18 · ⚛️ physics.chem-ph · physics.atom-ph

Free-particle Green's function matrix elements over spherical Gaussian and plane-wave-modulated Gaussian basis functions

Dibyendu Mahato , Wojciech Skomorowski This is my paper

Pith reviewed 2026-05-20 07:53 UTC · model grok-4.3

classification ⚛️ physics.chem-ph physics.atom-ph

keywords free-particle Green's functionGaussian basis functionsmatrix elementselectron scatteringFourier transformharmonic polynomialsrecurrence relationsasymptotic behavior

0 comments

The pith

A Fourier-based framework yields closed-form expressions for free-particle Green's function matrix elements in spherical Gaussian bases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a general analytical method to compute one- and two-center matrix elements of the free-particle Green's function over spherical Gaussian basis functions and plane-wave-modulated spherical Gaussians. This matters for electron scattering and autoionization studies because Gaussian bases are increasingly used for unbound states yet previously lacked efficient integral formulas. The derivation proceeds from the Fourier transform of the Gaussian functions combined with the addition theorem of harmonic polynomials. The resulting expressions are compact and closed-form, accompanied by recurrence relations for numerical efficiency. The work also derives the asymptotic behavior of these matrix elements, which is required when representing low-energy continuum electrons with finite bases.

Core claim

The central claim is that the free-particle Green's function matrix elements over spherical Gaussians and plane-wave-modulated Gaussians admit compact closed-form expressions and efficient recurrence relations obtained by taking the Fourier transform of the basis functions and invoking the addition theorem for harmonic polynomials. The same approach supplies the asymptotic form of the matrix elements at large distances, which is essential for continuum applications.

What carries the argument

Fourier transform of Gaussian functions together with the addition theorem of harmonic polynomials, which converts the integral into a form that separates into closed expressions and recurrences.

If this is right

Gaussian basis expansions become practical for calculating unbound and metastable electronic states in scattering theory.
Autoionization and electron-molecule scattering simulations can use finite Gaussian sets without resorting to numerical integration of the Green's function.
Asymptotic analysis supports accurate representation of low-energy continuum electrons inside a finite basis.
Recurrence relations reduce the computational cost of evaluating large numbers of such matrix elements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same transform-and-addition approach may generalize to matrix elements of other nonlocal operators in Gaussian bases.
Implementation in existing quantum-chemistry packages would immediately enable scattering calculations on polyatomic targets.
Numerical stability of the recurrences could be tested explicitly at high angular momentum or very diffuse exponents.

Load-bearing premise

The Fourier transform and addition theorem produce compact closed-form expressions and recurrence relations that remain numerically stable for the distances and exponents encountered in scattering calculations.

What would settle it

Direct numerical quadrature of the Green's function integral for chosen Gaussian centers, exponents, and angular momenta, followed by comparison to the analytic formula to machine precision.

Figures

Figures reproduced from arXiv: 2605.18564 by Dibyendu Mahato, Wojciech Skomorowski.

**Figure 2.** Figure 2: FIG. 2. One-center free-particle Green’s function matrix element [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Two-center free-particle Green’s function matrix element [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

read the original abstract

Free-particle Green's function plays a central role in the theoretical description of electron scattering and autoionization processes in quantum physics and chemistry. Recently, Gaussian basis set approaches have become increasingly important in applications to unbound and metastable electronic states. However, the practical use of such methods has been limited by the lack of efficient and compact analytical expressions for matrix elements of the free-particle Green's function in Gaussian-based representations. Here we present a novel, general analytical framework for the evaluation of one- and two-center matrix elements of the free-particle Green's function over spherical Gaussian basis functions and plane-wave-modulated spherical Gaussians. The derivation is based on the Fourier transform of Gaussian functions together with the addition theorem of harmonic polynomials, leading to compact closed-form expressions and efficient recurrence relations. We also analyze the asymptotic behavior of the free-particle Green's function matrix elements, which is essential in the description of low-energy continuum electrons using finite Gaussian basis sets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives closed-form matrix elements for the free-particle Green's function in Gaussian bases, a targeted advance for scattering calculations whose numerical stability still needs checking.

read the letter

The key takeaway is that this work gives a new set of analytical expressions for one- and two-center matrix elements of the free-particle Green's function using spherical Gaussian basis functions and plane-wave-modulated ones. The approach combines the Fourier transform of the Gaussians with the addition theorem for harmonic polynomials to arrive at compact closed forms and recurrence relations, plus some asymptotic analysis for low-energy cases.

Referee Report

1 major / 0 minor

Summary. The manuscript presents a novel analytical framework for evaluating one- and two-center matrix elements of the free-particle Green's function over spherical Gaussian basis functions and plane-wave-modulated spherical Gaussians. The approach is based on the Fourier transform of Gaussian functions combined with the addition theorem of harmonic polynomials, yielding compact closed-form expressions and efficient recurrence relations. The work also examines the asymptotic behavior of these matrix elements for applications to low-energy continuum electrons in finite Gaussian basis sets.

Significance. If the derivations are correct and the recurrence relations are numerically robust, the framework would address a longstanding practical barrier to using Gaussian basis sets for unbound and metastable states in scattering and autoionization calculations. It offers the potential for parameter-free, analytically compact evaluations that could improve efficiency and accuracy in quantum chemistry and physics applications involving continuum electrons.

major comments (1)

[Recurrence relations derivation and asymptotic analysis] The central claim that the Fourier-transform-plus-addition-theorem route produces recurrence relations that remain stable and efficient for small Gaussian exponents, small k, and large inter-center distances is load-bearing for the stated applicability to scattering calculations, yet the manuscript supplies no error analysis, numerical benchmarks, or tests for cancellation/overflow behavior in these regimes (see the section deriving the recurrence relations and the discussion of asymptotic behavior).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment. We address the point below and will revise the manuscript accordingly to strengthen the presentation of numerical robustness.

read point-by-point responses

Referee: [Recurrence relations derivation and asymptotic analysis] The central claim that the Fourier-transform-plus-addition-theorem route produces recurrence relations that remain stable and efficient for small Gaussian exponents, small k, and large inter-center distances is load-bearing for the stated applicability to scattering calculations, yet the manuscript supplies no error analysis, numerical benchmarks, or tests for cancellation/overflow behavior in these regimes (see the section deriving the recurrence relations and the discussion of asymptotic behavior).

Authors: We agree that explicit numerical validation would better support the applicability claims for scattering calculations. The manuscript derives the closed-form expressions and recurrence relations analytically and examines their asymptotic behavior for large inter-center distances and low-energy regimes. However, it does not include dedicated error analysis or benchmarks against direct numerical integration in the challenging limits of small Gaussian exponents and small k. In the revised manuscript we will add a dedicated subsection with numerical tests, including comparisons to quadrature results, forward and backward recurrence stability checks, and monitoring of cancellation/overflow for representative values of small exponents, small k, and large separations. These additions will quantify the practical efficiency and robustness of the relations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard external transforms and theorems

full rationale

The paper's central derivation applies the Fourier transform to Gaussian functions combined with the addition theorem for harmonic polynomials to obtain closed-form matrix elements and recurrence relations. These are established mathematical results from the broader literature, not defined in terms of the target Green's function matrix elements. No steps reduce by construction to fitted inputs, self-referential definitions, or load-bearing self-citations that lack independent verification. The expressions are presented as first-principles analytic results, with asymptotic analysis provided separately. This qualifies as a self-contained derivation against external benchmarks, consistent with a score of 0.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical identities rather than new physical postulates or fitted parameters.

axioms (2)

standard math Fourier transform of Gaussian functions converts position-space integrals to momentum space
Invoked in the derivation of the matrix elements as described in the abstract.
standard math Addition theorem of harmonic polynomials handles angular dependence
Used to obtain compact expressions for the angular parts of the integrals.

pith-pipeline@v0.9.0 · 5690 in / 1395 out tokens · 54198 ms · 2026-05-20T07:53:59.335086+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages

[1]

one ofµ a orµ b is zero (a) ifµ̸= 0 andµ a ̸= 0 andµ b = 0 ⟨lµ|laµa|lbµb⟩= 2⟨lµ a|laµa|lb0⟩ℜ [U µ lµa ]∗U µa laµa ,(A8) (b) ifµ̸= 0 andµ a = 0 andµ b ̸= 0 ⟨lµ|laµa|lbµb⟩= 2⟨lµ a|la0|lbµb⟩ℜ [U µ lµb ]∗U µb lbµb .(A9)

work page
[2]

(A12) The definition in Eq

ifµ a =µ b = 0 ⟨lµ|la0|lb0⟩=δ µ0⟨l0|la0|lb0⟩.(A10) The above expressions involve the unitary transformation matrixU µ lm, defined as U µ lm =δ m0δµ0 + 1√ 2 Θ(µ)δmµ + Θ(−µ)(+i)(−1)mδmµ +Θ(−µ)(−i)δm−µ + Θ(µ)(−1)mδm−µ ,(A11) whereδ mµ denotes the Kronecker delta and Θ(µ) = ( 1 forµ >0, 0 forµ≤0. (A12) The definition in Eq. (A12) ensures thatU µ lm = 0 for|µ|...

work page
[3]

Complex solid spherical harmonics Addition theorem for the solid spherical harmonics [40, 41]: Ylm(r1 +r 2) = 4π lX l′=0 min(l′,m+l−l′)X m′ =max (−l′ ,m−l+l′ ) G(lm|l′m′)Yl′m′(r1)Yl−l′,m−m′(r2),(D1) whereG(lm|l ′m′) coefficients are defined as: G(lm|l′m′) = " 2l+ 1 4π(2l′ + 1)[2(l−l ′) + 1] l+m l′ +m ′ l−m l′ −m ′ #1/2 .(D2)

work page
[4]

(56) and (57)

Real solid spherical harmonics The addition theorem for unnormalized real regular solid harmonics can be in general expressed as [32], zµ l (r1 +r 2) = lX l′=0 l′ X µ′=−l′ zµ′ l′ (r1) X µ′′ zµ′′ l−l′(r2)clµ l′µ′µ′′ ,(D3) where the unnormalized real harmonicz µ l (r) is defined in Eq. (56) and (57). To provide explicit expressions for the coefficientsc lµ ...

work page
[5]

forµ≥0 zµ l (r1 +r 2) = lX l′=0 min(l′,µ)X µ′ =max (0,l′−l+µ) Alµ l′µ′ h zµ′ l′ (r1)zµ−µ′ l−l′ (r2) −(1−δ µ′,0)(1−δ µ′,µ)z−µ′ l′ (r1)z−(µ−µ′) l−l′ (r2) i + l−1X l′=µ+1 min(l′,l−′+µ)X µ′=µ+1 Blµ l′µ′ h zµ′ l′ (r1)zµ′−µ l−l′ (r2) +z−µ′ l′ (r1)z−(µ′−µ) l−l′ (r2) i + l−µ−1X l′=1 min(l′,l−l′−µ)X µ′=1 C lµ l′µ′ h zµ′ l′ (r1)zµ′+µ l−l′ (r2) +z−µ′ l′ (r1)z−(µ′+µ)...

work page
[6]

forµ <0 z−|µ| l (r1 +r 2) = lX l′=0 min(l′,|µ|)X µ′ =max (0,l′ −l+|µ|) Al|µ| l′µ′ h (1−δ µ′,|µ|)zµ′ l′ (r1)z−(|µ|−µ′) l−l′ (r2) +(1−δ µ′,0)z−µ′ l′ (r1)z(|µ|−µ′) l−l′ (r2) i + l−1X l′=|µ|+1 min(l′,l−l′+|µ|)X µ′=|µ|+1 Bl|µ| l′µ′ h −z µ′ l′ (r1)z−(µ′−|µ|) l−l′ (r2) +z−µ′ l′ (r1)z(µ′−|µ|) l−l′ (r2) i + l−|µ|−1X l′=1 min(l′,l−l′−|µ|)X µ′=1 C l|µ| l′µ′ h zµ′ l′...

work page
[7]

forµ≥0 eik·rϕµ l (α,r−C) =N l(α)N ′ lµNk(α) lX l′=0 min(l′,µ)X µ′ =max (0,l′ −l+µ) Al,µ l′,µ′ Nl′(α) h ϕµ′ l′ (α,r−C †) N ′ l′µ′ zµ−µ′ l−l′ ik 2α −(1−δ µ′,0)(1−δ µ′,µ) ϕ−µ′ l′ (α,r−C †) N ′ l′−µ′ z−(µ−µ′) l−l′ ik 2α i + l−1X l′=µ+1 min(l′,l−l′+µ)X µ′=µ+1 Bl,µ l′,µ′ Nl′(α) h ϕµ′ l′ (α,r−C †) N ′ l′µ′ zµ′−µ l−l′ ik 2α + ϕ−µ′ l′ (α,r−C †) N ′ l′−µ′ z−(µ′−µ) ...

work page
[8]

(D9) and (D10) can be easily transformed into momentum space by Fourier transforming the functionsϕ µ′ l′ (α,r−C †), since all remaining terms are independent ofr

forµ <0 eik·rϕ−|µ| l (α,r−C) =N l(α)N ′ lµNk(α) lX l′=0 min(l′,|µ|)X µ′=max (0,l′ −l+|µ|) Al,|µ| l′,µ′ Nl′(α) h (1−δ µ′,|µ|) ϕµ′ l′ (α,r−C †) N ′ l′µ′ ×z−(|µ|−µ′) l−l′ ik 2α + (1−δ µ′,0) ϕ−µ′ l′ (α,r−C †) N ′ l′−µ′ z(|µ|−µ′) l−l′ ik 2α i + l−1X l′=|µ|+1 min(l′,l−l′+|µ|)X µ′=|µ|+1 Bl,|µ| l′,µ′ Nl′(α) h − ϕµ′ l′ (α,r−C †) N ′ l′µ′ z−(µ′−|µ|) l−l′ (r2) + ϕ−µ...

work page
[9]

forµ a ≥0 andµ b ≥0 Olbµb laµa (α, β,R †,k 1,k 2) =D lbµb laµa (α, β) 3X i,j=1 ⟨a(i)(α,A †,k 1)|bO|a(j)(β,B †,k 2)⟩,(D11)

work page
[10]

forµ a ≥0 andµ b <0 Olbµb laµa (α, β,R †,k 1,k 2) =D lbµb laµa (α, β) 3X i,j=1 ⟨a(i)(α,A †,k 1)|bO|b(j)(β,B †,k 2)⟩,(D12)

work page
[11]

forµ a <0 andµ b <0 Olbµb laµa (α, β,R †,k 1,k 2) =D lbµb laµa (α, β) 3X i,j=1 ⟨b(i)(α,A †,k 1)|bO|b(j)(β,B †,k 2)⟩,(D13)

work page
[12]

(D9) and (D10) into the general expression yields the complete matrix-element formulas given in Eqs

forµ a <0 andµ b ≥0 Olbµb laµa (α, β,R †,k 1,k 2) =D lbµb laµa (α, β) 3X i,j=1 ⟨b(i)(α,A †,k 1)|bO|a(j)(β,B †,k 2)⟩.(D14) Substituting the expansions from Eqs. (D9) and (D10) into the general expression yields the complete matrix-element formulas given in Eqs. (D11)–(D14). Each matrix element consists of a sum of nine terms. For case 1, the first term 19 ...

work page
[13]

S. F. Boys and A. C. Egerton, Electronic wave functions - I. A general method of calculation for the stationary states of any molecular system, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences200, 542 (1950)

work page 1950
[14]

Moiseyev,Non-Hermitian quantum mechanics(Cam- bridge University Press, 2011)

N. Moiseyev,Non-Hermitian quantum mechanics(Cam- bridge University Press, 2011)

work page 2011
[15]

Riss and H.-D

U. Riss and H.-D. Meyer, Calculation of resonance en- ergies and widths using the complex absorbing potential method, Journal of Physics B: Atomic, Molecular and Optical Physics26, 4503 (1993)

work page 1993
[16]

Moiseyev and C

N. Moiseyev and C. Corcoran, Autoionizing states of H2 and H2- using the complex-scaling method, Physical Re- view A20, 814 (1979)

work page 1979
[17]

McCurdy Jr and T

C. McCurdy Jr and T. Rescigno, Extension of the method of complex basis functions to molecular resonances, Phys- ical Review Letters41, 1364 (1978)

work page 1978
[18]

Moiseyev, Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling, Physics Reports302, 212 (1998)

N. Moiseyev, Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling, Physics Reports302, 212 (1998)

work page 1998
[19]

Jagau, K

T.-C. Jagau, K. B. Bravaya, and A. I. Krylov, Extending quantum chemistry of bound states to electronic reso- nances, Annual Review of Physical Chemistry68, 525 (2017)

work page 2017
[20]

Rescigno, C

T. Rescigno, C. McCurdy, and V. McKoy, Discrete ba- sis set approach to nonspherical scattering, Chemical Physics Letters27, 401 (1974)

work page 1974
[21]

Fliflet, D

A. Fliflet, D. Levin, M. Ma, and V. McKoy, Discrete- basis-set calculation for e-N2 scattering cross sections in the static-exchange approximation, Physical Review A 17, 160 (1978)

work page 1978
[22]

D. K. Watson and V. McKoy, Discrete-basis-function ap- proach to electron-molecule scattering, Physical Review A20, 1474 (1979)

work page 1979
[23]

Kaufmann, W

K. Kaufmann, W. Baumeister, and M. Jungen, Univer- sal Gaussian basis sets for an optimum representation of Rydberg and continuum wavefunctions, Journal of Physics B: Atomic, Molecular and Optical Physics22, 2223 (1989)

work page 1989
[24]

B. M. Nestmann and S. D. Peyerimhoff, Optimized gaus- sian basis sets for representation of continuum wavefunc- tions, Journal of Physics B: Atomic, Molecular and Op- tical Physics23, L773 (1990)

work page 1990
[25]

Faure, J

A. Faure, J. D. Gorfinkiel, L. A. Morgan, and J. Ten- nyson, GTOBAS: fitting continuum functions with Gaussian-type orbitals, Computer Physics Communica- tions144, 224 (2002)

work page 2002
[26]

Ammar, L

A. Ammar, L. U. Ancarani, and A. Leclerc, A complex Gaussian approach to molecular photoionization, Journal of Computational Chemistry42, 2294 (2021)

work page 2021
[27]

C. A. Moyer, On the Green function for a particle in a uniform electric field, Journal of Physics C: Solid State 20 Physics6, 1461 (1973)

work page 1973
[28]

Greenman, R

L. Greenman, R. R. Lucchese, and C. W. McCurdy, Vari- ational treatment of electron–polyatomic-molecule scat- tering calculations using adaptive overset grids, Phys. Rev. A96, 052706 (2017)

work page 2017
[29]

B. M. Nestmann, V. Krumbach, and S. D. Peyerimhoff, Numerical approach to energy-dependent complex- potential surfaces of metastable negative molecular ions, Phys. Rev. A42, 5406 (1990)

work page 1990
[30]

Ostlund, Polyatomic scattering integrals with Gaus- sian orbitals, Chemical Physics Letters34, 419 (1975)

N. Ostlund, Polyatomic scattering integrals with Gaus- sian orbitals, Chemical Physics Letters34, 419 (1975)

work page 1975
[31]

D. A. Levin, A. W. Fliflet, M. Ma, and V. McKoy, Gaus- sian matrix elements of the free-particle Green’s function, Journal of Computational Physics28, 416 (1978)

work page 1978
[32]

ˇC´ arsky, V

P. ˇC´ arsky, V. Hrouda, and M. Pol´ aˇ sek, New general for- mulas for matrix elements of the free-particle Green’s function over Cartesian Gaussians, Theoretica Chimica Acta93, 49 (1996)

work page 1996
[33]

Fiori and J

M. Fiori and J. Miraglia, New approach for approximat- ing the continuum wave function by Gaussian basis set, Computer Physics Communications183, 2528 (2012)

work page 2012
[34]

Bubin and L

S. Bubin and L. Adamowicz, Computer program ATOM- MOL-nonBO for performing calculations of ground and excited states of atoms and molecules without assuming the Born–Oppenheimer approximation using all-particle complex explicitly correlated Gaussian functions, The Journal of Chemical Physics152, 204102 (2020)

work page 2020
[35]

Ammar, A

A. Ammar, A. Leclerc, and L. U. Ancarani, Chapter four- teen - multicenter integrals involving complex Gaussian- type functions, inNew Electron Correlation Methods and their Applications, and Use of Atomic Orbitals with Ex- ponential Asymptotes, Advances in Quantum Chemistry, Vol. 83, edited by M. Musial and P. E. Hoggan (Academic Press, 2021) pp. 287–304

work page 2021
[36]

Epifanovsky, A

E. Epifanovsky, A. T. Gilbert, X. Feng, J. Lee, Y. Mao, N. Mardirossian, P. Pokhilko, A. F. White, M. P. Coons, A. L. Dempwolff,et al., Software for the frontiers of quan- tum chemistry: An overview of developments in the q- chem 5 package, The Journal of chemical physics155 (2021)

work page 2021
[37]

J. D. Louck,Angular momentum in quantum physics: theory and application(Addison-Wesley, 1981)

work page 1981
[38]

E. U. Condon and G. H. Shortley,The theory of atomic spectra(Cambridge University Press, 1935)

work page 1935
[39]

Kuang and C

J. Kuang and C. D. Lin, Molecular integrals over spher- ical Gaussian-type orbitals: I, Journal of Physics B: Atomic, Molecular and Optical Physics30, 2529 (1997)

work page 1997
[40]

B. Gao, A. J. Thorvaldsen, and K. Ruud, Gen1int: A unified procedure for the evaluation of one-electron in- tegrals over gaussian basis functions and their geometric derivatives, International Journal of Quantum Chemistry 111, 858 (2011)

work page 2011
[41]

Colle, A

R. Colle, A. Fortunelli, and S. Simonucci, Hermite Gaus- sian functions modulated by plane waves: a general basis set for bound and continuum states, Il Nuovo Cimento D 10, 805 (1988)

work page 1988
[42]

Allison, N

D. Allison, N. Handy, and S. Boys, A new basis set for molecular wavefunctions, Molecular Physics26, 715 (1973)

work page 1973
[43]

Dose and C

V. Dose and C. Semini, Coupled state calculation of pro- ton hydrogen collisions in a Gaussian basis, Helvetica Physica Acta47(1974)

work page 1974
[44]

J. F. Rico, R. L´ opez, I. Ema, and G. Ram´ ırez, Translation of real solid spherical harmonics, International Journal of Quantum Chemistry113, 1544 (2013)

work page 2013
[45]

W. R. Inc., Mathematica, Version 13.3 (2023), cham- paign, IL

work page 2023
[46]

H. B. Schlegel and M. J. Frisch, Transformation between cartesian and pure spherical harmonic gaussians, Inter- national Journal of Quantum Chemistry54, 83 (1995)

work page 1995
[47]

Ribaldone and J

C. Ribaldone and J. K. Desmarais, Spherical to Cartesian coordinates transformation for solid harmonics revisited, 2412.16733 (2024)

work page arXiv 2024
[48]

A. R. Edmonds,Angular momentum in quantum me- chanics, Vol. 4 (Princeton university press, 1996)

work page 1996
[49]

H. H. Homeier and E. Steinborn, Some properties of the coupling coefficients of real spherical harmonics and their relation to Gaunt coefficients, Journal of Molecular Structure: THEOCHEM368, 31 (1996)

work page 1996
[50]

Biedenharn and J

L. Biedenharn and J. Louck,Angular Momentum in Quantum Physics: Theory and Application, Angular Momentum in Quantum Physics (Cambridge University Press, 1984)

work page 1984
[51]

Kuang and C

J. Kuang and C. D. Lin, Molecular integrals over spheri- cal Gaussian-type orbitals: II. Modified with plane-wave phase factors, Journal of Physics B: Atomic, Molecular and Optical Physics30, 2549 (1997)

work page 1997
[52]

E. O. Steinborn and K. Ruedenberg, Rotation and trans- lation of regular and irregular solid spherical harmonics, Advances in Quantum Chemistry7, 1 (1973)

work page 1973
[53]

H. H. Homeier and E. Otto Steinborn, Improved quadra- ture methods for three-center nuclear attraction inte- grals with exponential-type basis functions, International Journal of Quantum Chemistry39, 625 (1991)

work page 1991

[1] [1]

one ofµ a orµ b is zero (a) ifµ̸= 0 andµ a ̸= 0 andµ b = 0 ⟨lµ|laµa|lbµb⟩= 2⟨lµ a|laµa|lb0⟩ℜ [U µ lµa ]∗U µa laµa ,(A8) (b) ifµ̸= 0 andµ a = 0 andµ b ̸= 0 ⟨lµ|laµa|lbµb⟩= 2⟨lµ a|la0|lbµb⟩ℜ [U µ lµb ]∗U µb lbµb .(A9)

work page

[2] [2]

(A12) The definition in Eq

ifµ a =µ b = 0 ⟨lµ|la0|lb0⟩=δ µ0⟨l0|la0|lb0⟩.(A10) The above expressions involve the unitary transformation matrixU µ lm, defined as U µ lm =δ m0δµ0 + 1√ 2 Θ(µ)δmµ + Θ(−µ)(+i)(−1)mδmµ +Θ(−µ)(−i)δm−µ + Θ(µ)(−1)mδm−µ ,(A11) whereδ mµ denotes the Kronecker delta and Θ(µ) = ( 1 forµ >0, 0 forµ≤0. (A12) The definition in Eq. (A12) ensures thatU µ lm = 0 for|µ|...

work page

[3] [3]

Complex solid spherical harmonics Addition theorem for the solid spherical harmonics [40, 41]: Ylm(r1 +r 2) = 4π lX l′=0 min(l′,m+l−l′)X m′ =max (−l′ ,m−l+l′ ) G(lm|l′m′)Yl′m′(r1)Yl−l′,m−m′(r2),(D1) whereG(lm|l ′m′) coefficients are defined as: G(lm|l′m′) = " 2l+ 1 4π(2l′ + 1)[2(l−l ′) + 1] l+m l′ +m ′ l−m l′ −m ′ #1/2 .(D2)

work page

[4] [4]

(56) and (57)

Real solid spherical harmonics The addition theorem for unnormalized real regular solid harmonics can be in general expressed as [32], zµ l (r1 +r 2) = lX l′=0 l′ X µ′=−l′ zµ′ l′ (r1) X µ′′ zµ′′ l−l′(r2)clµ l′µ′µ′′ ,(D3) where the unnormalized real harmonicz µ l (r) is defined in Eq. (56) and (57). To provide explicit expressions for the coefficientsc lµ ...

work page

[5] [5]

forµ≥0 zµ l (r1 +r 2) = lX l′=0 min(l′,µ)X µ′ =max (0,l′−l+µ) Alµ l′µ′ h zµ′ l′ (r1)zµ−µ′ l−l′ (r2) −(1−δ µ′,0)(1−δ µ′,µ)z−µ′ l′ (r1)z−(µ−µ′) l−l′ (r2) i + l−1X l′=µ+1 min(l′,l−′+µ)X µ′=µ+1 Blµ l′µ′ h zµ′ l′ (r1)zµ′−µ l−l′ (r2) +z−µ′ l′ (r1)z−(µ′−µ) l−l′ (r2) i + l−µ−1X l′=1 min(l′,l−l′−µ)X µ′=1 C lµ l′µ′ h zµ′ l′ (r1)zµ′+µ l−l′ (r2) +z−µ′ l′ (r1)z−(µ′+µ)...

work page

[6] [6]

forµ <0 z−|µ| l (r1 +r 2) = lX l′=0 min(l′,|µ|)X µ′ =max (0,l′ −l+|µ|) Al|µ| l′µ′ h (1−δ µ′,|µ|)zµ′ l′ (r1)z−(|µ|−µ′) l−l′ (r2) +(1−δ µ′,0)z−µ′ l′ (r1)z(|µ|−µ′) l−l′ (r2) i + l−1X l′=|µ|+1 min(l′,l−l′+|µ|)X µ′=|µ|+1 Bl|µ| l′µ′ h −z µ′ l′ (r1)z−(µ′−|µ|) l−l′ (r2) +z−µ′ l′ (r1)z(µ′−|µ|) l−l′ (r2) i + l−|µ|−1X l′=1 min(l′,l−l′−|µ|)X µ′=1 C l|µ| l′µ′ h zµ′ l′...

work page

[7] [7]

forµ≥0 eik·rϕµ l (α,r−C) =N l(α)N ′ lµNk(α) lX l′=0 min(l′,µ)X µ′ =max (0,l′ −l+µ) Al,µ l′,µ′ Nl′(α) h ϕµ′ l′ (α,r−C †) N ′ l′µ′ zµ−µ′ l−l′ ik 2α −(1−δ µ′,0)(1−δ µ′,µ) ϕ−µ′ l′ (α,r−C †) N ′ l′−µ′ z−(µ−µ′) l−l′ ik 2α i + l−1X l′=µ+1 min(l′,l−l′+µ)X µ′=µ+1 Bl,µ l′,µ′ Nl′(α) h ϕµ′ l′ (α,r−C †) N ′ l′µ′ zµ′−µ l−l′ ik 2α + ϕ−µ′ l′ (α,r−C †) N ′ l′−µ′ z−(µ′−µ) ...

work page

[8] [8]

(D9) and (D10) can be easily transformed into momentum space by Fourier transforming the functionsϕ µ′ l′ (α,r−C †), since all remaining terms are independent ofr

forµ <0 eik·rϕ−|µ| l (α,r−C) =N l(α)N ′ lµNk(α) lX l′=0 min(l′,|µ|)X µ′=max (0,l′ −l+|µ|) Al,|µ| l′,µ′ Nl′(α) h (1−δ µ′,|µ|) ϕµ′ l′ (α,r−C †) N ′ l′µ′ ×z−(|µ|−µ′) l−l′ ik 2α + (1−δ µ′,0) ϕ−µ′ l′ (α,r−C †) N ′ l′−µ′ z(|µ|−µ′) l−l′ ik 2α i + l−1X l′=|µ|+1 min(l′,l−l′+|µ|)X µ′=|µ|+1 Bl,|µ| l′,µ′ Nl′(α) h − ϕµ′ l′ (α,r−C †) N ′ l′µ′ z−(µ′−|µ|) l−l′ (r2) + ϕ−µ...

work page

[9] [9]

forµ a ≥0 andµ b ≥0 Olbµb laµa (α, β,R †,k 1,k 2) =D lbµb laµa (α, β) 3X i,j=1 ⟨a(i)(α,A †,k 1)|bO|a(j)(β,B †,k 2)⟩,(D11)

work page

[10] [10]

forµ a ≥0 andµ b <0 Olbµb laµa (α, β,R †,k 1,k 2) =D lbµb laµa (α, β) 3X i,j=1 ⟨a(i)(α,A †,k 1)|bO|b(j)(β,B †,k 2)⟩,(D12)

work page

[11] [11]

forµ a <0 andµ b <0 Olbµb laµa (α, β,R †,k 1,k 2) =D lbµb laµa (α, β) 3X i,j=1 ⟨b(i)(α,A †,k 1)|bO|b(j)(β,B †,k 2)⟩,(D13)

work page

[12] [12]

(D9) and (D10) into the general expression yields the complete matrix-element formulas given in Eqs

forµ a <0 andµ b ≥0 Olbµb laµa (α, β,R †,k 1,k 2) =D lbµb laµa (α, β) 3X i,j=1 ⟨b(i)(α,A †,k 1)|bO|a(j)(β,B †,k 2)⟩.(D14) Substituting the expansions from Eqs. (D9) and (D10) into the general expression yields the complete matrix-element formulas given in Eqs. (D11)–(D14). Each matrix element consists of a sum of nine terms. For case 1, the first term 19 ...

work page

[13] [13]

S. F. Boys and A. C. Egerton, Electronic wave functions - I. A general method of calculation for the stationary states of any molecular system, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences200, 542 (1950)

work page 1950

[14] [14]

Moiseyev,Non-Hermitian quantum mechanics(Cam- bridge University Press, 2011)

N. Moiseyev,Non-Hermitian quantum mechanics(Cam- bridge University Press, 2011)

work page 2011

[15] [15]

Riss and H.-D

U. Riss and H.-D. Meyer, Calculation of resonance en- ergies and widths using the complex absorbing potential method, Journal of Physics B: Atomic, Molecular and Optical Physics26, 4503 (1993)

work page 1993

[16] [16]

Moiseyev and C

N. Moiseyev and C. Corcoran, Autoionizing states of H2 and H2- using the complex-scaling method, Physical Re- view A20, 814 (1979)

work page 1979

[17] [17]

McCurdy Jr and T

C. McCurdy Jr and T. Rescigno, Extension of the method of complex basis functions to molecular resonances, Phys- ical Review Letters41, 1364 (1978)

work page 1978

[18] [18]

Moiseyev, Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling, Physics Reports302, 212 (1998)

N. Moiseyev, Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling, Physics Reports302, 212 (1998)

work page 1998

[19] [19]

Jagau, K

T.-C. Jagau, K. B. Bravaya, and A. I. Krylov, Extending quantum chemistry of bound states to electronic reso- nances, Annual Review of Physical Chemistry68, 525 (2017)

work page 2017

[20] [20]

Rescigno, C

T. Rescigno, C. McCurdy, and V. McKoy, Discrete ba- sis set approach to nonspherical scattering, Chemical Physics Letters27, 401 (1974)

work page 1974

[21] [21]

Fliflet, D

A. Fliflet, D. Levin, M. Ma, and V. McKoy, Discrete- basis-set calculation for e-N2 scattering cross sections in the static-exchange approximation, Physical Review A 17, 160 (1978)

work page 1978

[22] [22]

D. K. Watson and V. McKoy, Discrete-basis-function ap- proach to electron-molecule scattering, Physical Review A20, 1474 (1979)

work page 1979

[23] [23]

Kaufmann, W

K. Kaufmann, W. Baumeister, and M. Jungen, Univer- sal Gaussian basis sets for an optimum representation of Rydberg and continuum wavefunctions, Journal of Physics B: Atomic, Molecular and Optical Physics22, 2223 (1989)

work page 1989

[24] [24]

B. M. Nestmann and S. D. Peyerimhoff, Optimized gaus- sian basis sets for representation of continuum wavefunc- tions, Journal of Physics B: Atomic, Molecular and Op- tical Physics23, L773 (1990)

work page 1990

[25] [25]

Faure, J

A. Faure, J. D. Gorfinkiel, L. A. Morgan, and J. Ten- nyson, GTOBAS: fitting continuum functions with Gaussian-type orbitals, Computer Physics Communica- tions144, 224 (2002)

work page 2002

[26] [26]

Ammar, L

A. Ammar, L. U. Ancarani, and A. Leclerc, A complex Gaussian approach to molecular photoionization, Journal of Computational Chemistry42, 2294 (2021)

work page 2021

[27] [27]

C. A. Moyer, On the Green function for a particle in a uniform electric field, Journal of Physics C: Solid State 20 Physics6, 1461 (1973)

work page 1973

[28] [28]

Greenman, R

L. Greenman, R. R. Lucchese, and C. W. McCurdy, Vari- ational treatment of electron–polyatomic-molecule scat- tering calculations using adaptive overset grids, Phys. Rev. A96, 052706 (2017)

work page 2017

[29] [29]

B. M. Nestmann, V. Krumbach, and S. D. Peyerimhoff, Numerical approach to energy-dependent complex- potential surfaces of metastable negative molecular ions, Phys. Rev. A42, 5406 (1990)

work page 1990

[30] [30]

Ostlund, Polyatomic scattering integrals with Gaus- sian orbitals, Chemical Physics Letters34, 419 (1975)

N. Ostlund, Polyatomic scattering integrals with Gaus- sian orbitals, Chemical Physics Letters34, 419 (1975)

work page 1975

[31] [31]

D. A. Levin, A. W. Fliflet, M. Ma, and V. McKoy, Gaus- sian matrix elements of the free-particle Green’s function, Journal of Computational Physics28, 416 (1978)

work page 1978

[32] [32]

ˇC´ arsky, V

P. ˇC´ arsky, V. Hrouda, and M. Pol´ aˇ sek, New general for- mulas for matrix elements of the free-particle Green’s function over Cartesian Gaussians, Theoretica Chimica Acta93, 49 (1996)

work page 1996

[33] [33]

Fiori and J

M. Fiori and J. Miraglia, New approach for approximat- ing the continuum wave function by Gaussian basis set, Computer Physics Communications183, 2528 (2012)

work page 2012

[34] [34]

Bubin and L

S. Bubin and L. Adamowicz, Computer program ATOM- MOL-nonBO for performing calculations of ground and excited states of atoms and molecules without assuming the Born–Oppenheimer approximation using all-particle complex explicitly correlated Gaussian functions, The Journal of Chemical Physics152, 204102 (2020)

work page 2020

[35] [35]

Ammar, A

A. Ammar, A. Leclerc, and L. U. Ancarani, Chapter four- teen - multicenter integrals involving complex Gaussian- type functions, inNew Electron Correlation Methods and their Applications, and Use of Atomic Orbitals with Ex- ponential Asymptotes, Advances in Quantum Chemistry, Vol. 83, edited by M. Musial and P. E. Hoggan (Academic Press, 2021) pp. 287–304

work page 2021

[36] [36]

Epifanovsky, A

E. Epifanovsky, A. T. Gilbert, X. Feng, J. Lee, Y. Mao, N. Mardirossian, P. Pokhilko, A. F. White, M. P. Coons, A. L. Dempwolff,et al., Software for the frontiers of quan- tum chemistry: An overview of developments in the q- chem 5 package, The Journal of chemical physics155 (2021)

work page 2021

[37] [37]

J. D. Louck,Angular momentum in quantum physics: theory and application(Addison-Wesley, 1981)

work page 1981

[38] [38]

E. U. Condon and G. H. Shortley,The theory of atomic spectra(Cambridge University Press, 1935)

work page 1935

[39] [39]

Kuang and C

J. Kuang and C. D. Lin, Molecular integrals over spher- ical Gaussian-type orbitals: I, Journal of Physics B: Atomic, Molecular and Optical Physics30, 2529 (1997)

work page 1997

[40] [40]

B. Gao, A. J. Thorvaldsen, and K. Ruud, Gen1int: A unified procedure for the evaluation of one-electron in- tegrals over gaussian basis functions and their geometric derivatives, International Journal of Quantum Chemistry 111, 858 (2011)

work page 2011

[41] [41]

Colle, A

R. Colle, A. Fortunelli, and S. Simonucci, Hermite Gaus- sian functions modulated by plane waves: a general basis set for bound and continuum states, Il Nuovo Cimento D 10, 805 (1988)

work page 1988

[42] [42]

Allison, N

D. Allison, N. Handy, and S. Boys, A new basis set for molecular wavefunctions, Molecular Physics26, 715 (1973)

work page 1973

[43] [43]

Dose and C

V. Dose and C. Semini, Coupled state calculation of pro- ton hydrogen collisions in a Gaussian basis, Helvetica Physica Acta47(1974)

work page 1974

[44] [44]

J. F. Rico, R. L´ opez, I. Ema, and G. Ram´ ırez, Translation of real solid spherical harmonics, International Journal of Quantum Chemistry113, 1544 (2013)

work page 2013

[45] [45]

W. R. Inc., Mathematica, Version 13.3 (2023), cham- paign, IL

work page 2023

[46] [46]

H. B. Schlegel and M. J. Frisch, Transformation between cartesian and pure spherical harmonic gaussians, Inter- national Journal of Quantum Chemistry54, 83 (1995)

work page 1995

[47] [47]

Ribaldone and J

C. Ribaldone and J. K. Desmarais, Spherical to Cartesian coordinates transformation for solid harmonics revisited, 2412.16733 (2024)

work page arXiv 2024

[48] [48]

A. R. Edmonds,Angular momentum in quantum me- chanics, Vol. 4 (Princeton university press, 1996)

work page 1996

[49] [49]

H. H. Homeier and E. Steinborn, Some properties of the coupling coefficients of real spherical harmonics and their relation to Gaunt coefficients, Journal of Molecular Structure: THEOCHEM368, 31 (1996)

work page 1996

[50] [50]

Biedenharn and J

L. Biedenharn and J. Louck,Angular Momentum in Quantum Physics: Theory and Application, Angular Momentum in Quantum Physics (Cambridge University Press, 1984)

work page 1984

[51] [51]

Kuang and C

J. Kuang and C. D. Lin, Molecular integrals over spheri- cal Gaussian-type orbitals: II. Modified with plane-wave phase factors, Journal of Physics B: Atomic, Molecular and Optical Physics30, 2549 (1997)

work page 1997

[52] [52]

E. O. Steinborn and K. Ruedenberg, Rotation and trans- lation of regular and irregular solid spherical harmonics, Advances in Quantum Chemistry7, 1 (1973)

work page 1973

[53] [53]

H. H. Homeier and E. Otto Steinborn, Improved quadra- ture methods for three-center nuclear attraction inte- grals with exponential-type basis functions, International Journal of Quantum Chemistry39, 625 (1991)

work page 1991