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Smeared Coulomb potential orbitals: I- Asymptotic
expansion
Patrick Cassam-Chenaï, Gilles Lebeau
To cite this version:
Patrick Cassam-Chenaï, Gilles Lebeau. Smeared Coulomb potential orbitals: I- Asymptotic expansion. Journal of Mathematical Chemistry, Springer Verlag (Germany), 2021, �10.1007/s10910-021-01218-6�. �hal-01539172v5�
Smeared Coulomb potential orbitals:
I- Asymptotic expansion
Patrick Cassam-Chena¨ı
∗and Gilles Lebeau
Universit´e Cˆote d’Azur, LJAD, UMR 7351, 06100 Nice, France E-mail: [email protected]
Abstract
We consider an 1-electron model Hamiltonian, whose potential energy corresponds to the Coulomb potential of an infinite wire with charge Z distributed according to a
Gaussian function. The time independent Schr¨odinger equation for this Hamiltonian is
solved perturbationally in the asymptotic limit of small amplitude vibration (Gaussian function width close to zero). We propose to use the naturally polarized functions so-obtained, as orbital basis sets for quantum chemical calculations. In particular, they should be well suited to perform electron-nucleus mean field configuration interaction calculations. Since the free-parameters of the model have the remarkable property to factorize the perturbative corrections to the eigenfunctions, these corrective part in factor can be simply added as additional functions to standard basis sets, leaving it to the molecular orbital calculation to optimize the free parameters within molecular orbital coefficients.
1
Introduction
This paper is dedicated to Prof. Graham Chandler for his 80thbirthday. His famous “Mclean
and Chandler basis sets” have proved extremely useful to the quantum chemistry community. Some twenty years ago, we optimized Gaussian basis sets for molecular fragments together with D. Jayatilaka and G. S. Chandler1. However, this endeavour was suspended due to technical difficulties raised by electronegative atoms. In the present article, we come back to, perhaps, a more original approach to basis functions, where the latter are not selected on the ground of their technical advantages, as was the case initially for Gaussian-type orbitals (GTO)2, but because they are eigenfunctions of a model Hamiltonian and therefore have
some physical relevance. It is hoped that this property can be taken advantage of in basis set truncation.
The model Hamiltonian we will consider, is a generalization of the hydrogenoid atom Coulomb Hamiltonian. So, our new orbitals will be part of the exponential-type orbitals (ETO)3 family, like the hydrogenoid atom eigenfunctions, which constitute an asymptotic
limit. Slater-type orbitals (STO)4,5 is another type of ETO related to hydrogenoid orbitals
(HO): they can be seen as “uncontracted” HO, that is to say, as HO with the Laguerre poly-nomial prefactor replaced by a simple mopoly-nomial one. In contrast to GTO, the difficulty of computing multicenter integrals with STO has led to the introduction of more ETO family members such as Bessel-type orbitals (BTO)6 or Coulomb-Sturmian orbitals (CSO)7. The techniques developed for the latter8–10 will be equally relevant for integrals involving our
deformed hydrogenoid orbitals (DHO).
It is our take that DHO will be particularly useful for the recently developed electron-nucleus mean field configuration interaction (EN-MFCI) method11. The EN-MFCI method
affords one to obtain in a single calculation, the electronic and vibrational energy levels of a molecule, without making the “Born-Oppenheimer” (BO) approximation11,12. In contrast,
the traditional methods of Quantum Chemistry are set in the frame of this approximation. They describe electronic clouds of fixed nuclear configurations and make use of orbital basis sets centered on nuclear positions. The latter basis sets are not appropriate for EN-MFCI calculations and their discrepancies have been bypassed so far, only by adding off-centered orbitals. However, the addition of such functions introduces linear dependencies within the orbital set and spans virtual molecular orbitals of little relevance for the description of low energy wave functions of the molecule. So, it appears important to develop new orbital basis functions for the EN-MFCI method, able to describe the smeared electronic cloud of oscillating nuclei in a molecular system.
This article is the first part of a series aiming at deriving appropriate basis functions for EN-MFCI calculations. We propose to use the eigenfunctions of a one-electron model Hamiltonian corresponding to a Coulomb potential convoluted with a Gaussian function. The latter can be seen as a ground state vibrational eigenfunction for the nuclear internal motion in the harmonic approximation. In the next section, we show how such Hamiltonian can be expanded asymptotically near the infinitely small Gaussian width limit, where the hydrogenoid atom eigenfunctions are retrieved. Then, in section 3, we solve the eigenproblem for the first order corrected potential by means of the Rayleigh-Schr¨odinger perturbation theory. We tabulate the fist order corrected eigenfunctions and associated eigenvalues up to n = 7. In the last section, we provide details and perspectives on how these eigenfunctions can be employed in multi-electron, quantum chemical calculations. Numerical applications of EN-MFCI making use of these new functions is postponed to part two of this series.
2
Asymptotic expansion of a smeared Coulombic
Hamil-tonian
Let us consider the one-electron model Hamiltonian: H = −2µM + V (~r), with a potential of the form: V (~r) = −Zr a π +∞ Z −∞ exp[−az02] k~r − ~rz0k dz0, (1)
where, Z ∈ N∗, a ∈ R∗+, ~r = (x, y, z) and ~rz0 = (0, 0, z0) in Cartesian coordinates. When ~r is expressed in cylindrical coordinates, ~r = (ρ, φ, z), the potential depends only upon ρ and z,
V (ρ, z) = −Zr a π +∞ Z −∞ exp[−az2 0] pρ2+ (z − z 0)2 dz0. (2)
This potential corresponds to the Coulomb potential of an infinite wire with charge Z dis-tributed according to a Gaussian function. In the limiting case of a Gaussian function sharply peaking at the origin (a → +∞, Dirac distribution limit), the system will tends towards a point-charge Z concentrated at the origin and the hydrogenoid atom eigenfunctions will be recovered.
However, by taking a Gaussian width parameter of the order of magnitude of a nucleus vibra-tion amplitude, we will get basis funcvibra-tions corresponding to a Coulomb potential convoluted by a nuclear, vibrational, harmonic motion, that we may think particularly appropriate for EN-MFCI calculations.
Unfortunately, the Schr¨odinger equation for this potential is hard to solve because the whole z-axis is singular. So, we will restrict ourselves to the a → ∞ asymptotic limit, and expand
the potential V (ρ, z) as V (ρ, z) = −Z √ a π +∞ R 0 dλ √ λ +∞ R −∞ dz0 exp[−az02− λ(ρ2+ (z − z0)2)] = −Zpπa +∞ R 0 dλ √ λ(a+λ)exp[−λ(ρ 2+ z2)]exp[ λ2 a+λz2] = −Z√ π +∞ R 0 dλ 1 √ λ + −λ2+z2λ2 √ λa + λ3/2(3−12z2λ+4z4λ2) 8a2 + λ5/2(−15+90z2λ−60z4λ2+8z6λ3) 48a3 + o 1a 72 exp[−λ(ρ2+ z2)] (3)
Swaping the limits and setting r =pρ2+ z2, ρ = r × sin(θ), z = r × cos(θ), with θ ∈ [0, π],
the potential becomes
V (r, θ) = −Zr + (1−3cos(θ)
2)Z
4r3a −
3((3−30cos(θ)2+35cos(θ)4)Z)
32r5a2 +
15(5−105cos(θ)2+315cos(θ)4−231cos(θ)6)Z
128r7a3 + O 1 a 4 , (4)
where we recognize the hydrogenoid atom potential in the zeroth-order term,
V(0)(r, θ) = −Z
r. (5)
Now, at any order, the singularity is located at the single point r = 0. In the following, we will only consider the V (r, θ) potential truncated at first order,
V1(r, θ) := −
Z r +
(1 − 3cos(θ)2) Z
4r3a . (6)
Since this potential is not bounded from below, we will not attempt to solve the eigenproblem exactly. Instead, we apply Rayleigh-Schr¨odinger perturbation theory, to obtain the first order corrections to the hydrogenoid atom eigenstates.
3
Perturbationally corrected eigenstates
So, we consider the following Hamiltonian in spherical coordinates and atomic units
H = −1 2µ 1 r2 ∂ ∂rr 2 ∂ ∂r + 1 r2sin(θ) ∂ ∂θsin(θ) ∂ ∂θ + 1 r2sin(θ)2 ∂2 ∂φ2 + V1(r, θ), (7)
defined on the Hilbert space of square integrable functions whose scalar product is expressed as, hψ1|ψ2i := +∞ Z 0 r2dr π Z 0 sin(θ)dθ 2π Z 0 dφ ψ∗1(r, θ, φ)ψ2(r, θ, φ). (8)
A Galerkin-type approach similar to the one proposed in13, with spherical harmonics in place
of Chebychev basis functions could be considered to solve its eigenvalue problem. However, it is more practical to approach the eigenstates perturbationally, starting from the well-known solutions of time-independent Schr¨odinger equation for the hydrogenoid atom,
ψ(0)n,l,m(r, θ, φ) = Rn,l(r)Yl,m(θ, φ) (9) with Rn,l(r) = 2µZ n 32 s (n − l − 1)! 2n[(n + l)!]exp −µZr n 2µZr n l L2l+1n−l−1 2µZr n , (10)
where L2l+1n−l−1(x) denotes the generalized Laguerre polynomials, and Yl,m(θ, φ) the spherical
harmonics. We note that the perturbation operator,
V(1)(r, θ) := 1 a (1 − 3cos(θ)2) Z 4r3 . (11) is proportional to Y2,0(θ, φ), V(1)(r, θ) =r π 5 −Z a Y2,0(θ, φ) r3 . (12)
Given the following integral formula π Z 0 sin(θ)dθ 2π Z 0 dφ Yl1,m1(θ, φ)Yl2,m2(θ, φ)Yl3,m3(θ, φ) = r (2l1+ 1)(2l2+ 1)(2l3+ 1) 4π l1 l2 l3 0 0 0 l1 l2 l3 m1 m2 m3 , (13)
and the well-known relation for the conjugate of a spherical harmonic:
Yl,m∗ (θ, φ) = (−1)mYl,−m(θ, φ), (14)
we deduce that, for a given (n, l, m)-triplet of quantum numbers, the state ψn,l,m(0) can only be coupled at first order to states ψn(0)0,l0,m’s such that (i) l0 ≥ |m|, (ii) for l0 ∈ {|l − 2|, · · · , l + 2}, the 3-j symbol l0 l 2 0 0 0
is non zero, and, (iii) n
0 > l0. The allowed quantum number values
are summed up in Tab.1.
Table 1: List of hydrogenoid eigenstates coupled by the perturbation operator of Eq.(12) to a given hydrogenoid eigenstate (in terms of their associated quantum numbers).
Zero-order states states possibly coupled by first-order perturbation n > 0, l = 0, m = 0 (n0> 2, l0= 2, m0= 0)
n > 1, l = 1, m ∈ {−1, 0, 1} (n0> 1, l0= 1, m0= m); (n0> 3, l0= 3, m0= m) n > l ≥ 2, m ∈ {−l, −l + 1, l − 1, l} (n0> l, l0= l, m0= m); (n0> l + 2, l0= l + 2, m0= m)
n > l ≥ 2, m ∈ {−l + 2, · · · , l − 2} (n0> l − 2, l0= l − 2, m0= m); (n0> l, l0= l, m0= m); (n0> l + 2, l0= l + 2, m0= m)
A priori, the perturbation operator needs to be diagonalized first in each degenerate n-subspace. The first order correction to the unperturbed energies, (that is the energies of the hydrogenoid atom, En,l,m(0) = −µZ2n22 in hartree), are the eigenvalues of the matrix,
(hψn,l,m(0) |V(1)|ψn,l(0)0,m0i)(l,m),(l0,m0)= (δm,m0hψ(0)
n,l,m|V
(1)|ψ(0)
n,l0,mi)(l,m),(l0,m0). (15) However, this matrix is already diagonal: For m = m0 and l 6= l0 a non-zero integration on angular variables implies l = l0± 2, as we have seen. Suppose without loss of general-ity, that l = l0 + 2, the problem being symmetrical in l and l0. The generalized Laguerre polynomial L2l+1n−l−1 2µZrn = L2ln−l0+50−3 2µZrn can be expanded as a linear combination of gen-eralized Laguerre polynomial L2li 0+1 2µZrn with i ≤ n − l0 − 3. All these polynomials are orthogonal to L2ln−l0+10−1 2µZrn for the mesure 2µZrn
(2l0+1)
to the factor appearing in front of the generalized Laguerre polynomials in the radial integral hRn,l|r13|Rn,l0i.
So, for l = 0, there is no first order correction,
En,0,0 = −µZ
2
2n2, (16)
and for l > 0, the correction is,
En,l,m = E (0) n,l,m− r π 5 Z a hψn,l,m(0) |Y2,0(θ, φ) r3 |ψ (0) n,l,mi = −µZ 2 2n2 − r π 5 Z a Z+∞ 0 R2 n,l(r) r dr × (−1) m(l + 1 2) r 5 π l l 2 0 0 0 l l 2 −m m 0 = −µZ 2 2n2 − Z a × (−1)m(l + 1 2) l l 2 0 0 0 l l 2 −m m 0 +∞ Z 0 R2n,l(r) r dr. (17) As expected, the spherical symmetry is broken: an (l, |m|)-dependency is introduced in the perturbed eigenvalues. In Tab.2, we provide the first order eigenvalues, which can be useful for basis set truncation purposes. We note that, at first order, degeneracy is not completely lifted, as for example, E4,0,0 = E4,3,±2. If we factorize by the zero order energy, we see that
Table 2: First-order corrected energies (up to n=4). For every pairs, (n, l), the sum over m ∈ {−l, −l + 1, · · · , l − 1, l} of the first order corrections is zero. The spacing between the energies does not follow any scale, only the order between the levels is respected.
l = 0 l = 1 l = 2 l = 3 E4,1,±1= −µZ 2 32 + µ3Z4 1920a E4,2,±2= −µZ 2 32 + µ3Z4 6760a E4,3,±3= −µZ 2 32 + µ3Z4 16128a n = 4 E4,0,0= −µZ 2 32 E4,3,±2= − µZ2 32 E4,3,±1= −µZ 2 32 − µ3Z4 26880a E4,3,0= −µZ 2 32 − µ3Z4 20160a E4,2,±1= −µZ 2 32 − µ3Z4 13440a E4,2,0= −µZ 2 32 − µ3Z4 6760a E4,1,0= −µZ 2 32 − µ3Z4 960a ... E3,1,±1= −µZ 2 18 + µ3Z4 810a E3,2,±2= −µZ 2 18 + µ3Z4 2835a n = 3 E3,0,0= −µZ 2 18 E3,2,±1= −µZ 2 18 − µ3Z4 5670a E3,2,0= −µZ 2 18 − µ3Z4 2835a E3,1,0= −µZ 2 18 − µ3Z4 405a ... E2,1,±1= −µZ 2 8 + µ3Z4 240a n = 2 E2,0,0= −µZ 2 8 E2,1,0= −µZ 2 8 − µ3Z4 120a ... n = 1 E1,0,0= −µZ 2 2
ψn,l,m = ψ (0) n,l,m+ X (n0,l0)6=(n,l) −pπ 5 Z a hψ (0) n0,l0,m| Y2,0(θ,φ) r3 |ψ (0) n,l,mi En,l,m(0) − En(0)0,l0,m ψn(0)0,l0,m = ψn,l,m(0) + X (n0,l0)6=(n,l) − Z a +∞R 0 Rn0,l0(r)Rn,l(r) r dr ×(−1)m q (l0+1 2)(l + 1 2) l0 l 2 0 0 0 l0 l 2 −m m 0 −µZ2 2n2 + µZ2 2n02 ψn(0)0,l0,m = ψn,l,m(0) + X (n0,l0)6=(n,l) 2n02n2 aµZ(n02−n2)(−1)m q (l0+1 2)(l + 1 2) l0 l 2 0 0 0 l0 l 2 −m m 0 +∞ R 0 Rn0 ,l0(r)Rn,l(r) r drψ (0) n0,l0,m, (18)
the radial integral being proportional to µ3Z3 (because of the r13-term, as it appears when one makes the change of variables x = 2µZr), all the coupling coefficients in the expansion are proportional to µ2aZ2. Here also, the potentially divergent terms in the expansion, due to the degeneracy of the zeroth order eigenvalues for a given n in the denominators, can be
excluded since the corresponding numerators cancel out for the same reason that makes the first order matrix, Eq.(15), diagonal.
The first terms in the expansion of the lowest eigenfunctions are given in Tab. 3. Iso-density surfaces of the Iso-density functions corresponding to these eigenfunctions are displayed in Figs. 1 to 10. They shows isosurfaces belonging sometimes to different homology groups for different density values, which give an idea of the repartition of the electronic charge as it becomes more and more concentrated. The parameter c := µ × Z, which controls how diffuse the unperturbed hydrogenoid orbitals are, is set to 1. The parameter b := µ×Za , which tunes the intensity of the distorsion with respect to spherical symmetry, takes two values: b = 1 close to the no distorsion case (b = 0), and b = 32 for which the effect of the perturbed potential becomes clearly apparent. The perturbed 1s orbital, Fig. 1 for example, becomes elongated along the Coulomb potential spreading axis, as one expects. The variations for other orbitals are sometimes less intuitive.
Table 3: First-order corrected wave functions, ordered in increasing energy eigenvalue up to n=3 (in Appendix we provide a more comprehensive table up to n=7, “i-orbitals”).
n = 1 l = 0 m = 0 ψ1,0,0= ψ1,0,0(0) + √ 5 5aµZ P n0 >2 1 1− 1 n02 +∞ R 0 Rn0 ,2(r)R1,0(r) r drψ (0) n0 ,2,0 = ψ(0)1,0,0+µ2 Z2a √ 6 480ψ (0) 3,2,0+28125104 ψ (0) 4,2,0+5 √ 14 6804ψ (0) 5,2,0+5744 √ 21 12353145ψ (0) 6,2,0+3299 √ 21 8847360ψ (0) 7,2,0+ · · · n = 2 l = 0 m = 0 ψ2,0,0= ψ2,0,0(0) + 4 √ 5 5aµZ P n0 >2 1 1− 4 n02 +∞ R 0 Rn0 ,2(r)R2,0(r) r drψ (0) n0 ,2,0 = ψ(0)2,0,0−µ2 Z2 a 32√3 9375ψ (0) 3,2,0+ 2 √ 2 1215ψ (0) 4,2,0+ 3680 √ 7 7411887ψ (0) 5,2,0+ 29 √ 42 215040ψ (0) 6,2,0+ 63584 √ 42 645700815ψ (0) 7,2,0+ · · · n = 2 l = 1 m = 0 ψ2,1,0= ψ2,1,0(0) +5aµZ8 P n0 >2 1 1− 4 n02 +∞ R 0 Rn0 ,1(r)R2,1(r) r drψ (0) n0 ,1,0+ 12√21 35aµZ P n0 >3 1 1− 4 n02 +∞ R 0 Rn0 ,3(r)R2,1(r) r drψ (0) n0 ,3,0 = ψ(0)2,1,0+µ2 Z2a 3125192ψ(0)3,1,0+56 √ 10 6075 ψ (0) 4,1,0+8768 √ 5 1058841ψ (0) 5,1,0+201 √ 35 89600 ψ (0) 6,1,0+2926784 √ 14 1076168025ψ (0) 7,1,0+ · · · +µ2 Z2a 8√10 14175ψ (0) 4,3,0+ 512 √ 5 823543ψ (0) 5,3,0+123 √ 10 358400ψ (0) 6,3,0+ 541184 √ 3 1076168025ψ (0) 7,3,0+ · · · n = 2 l = 1 m = ±1 ψ2,1,±1= ψ2,1,±1(0) −5aµZ4 P n0 >2 1 1− 4 n02 +∞ R 0 Rn0 ,1(r)R2,1(r) r drψ (0) n0 ,1,±1 +12 √ 14 35aµZ P n0 >3 1 1− 4 n02 +∞ R 0 Rn0 ,3(r)R2,1(r) r drψ (0) n0 ,3,±1 = ψ(0)2,1,±1−µ2 Z2a 96 3125ψ (0) 3,1,±1+ 28√10 6075 ψ (0) 4,1,±1+ 4384√5 1058841ψ (0) 5,1,±1+ 201√35 179200ψ (0) 6,1,±1+ 1463392√14 1076168025ψ (0) 7,1,±1+ · · · +µ2 Z2a 16 √ 15 42525ψ (0) 4,3,±1+ 512 √ 30 2470629ψ (0) 5,3,±1+ 41 √ 15 179200ψ (0) 6,3,±1+ 541184 √ 2 1076168025ψ (0) 7,3,±1+ · · · n = 3 l = 0 m = 0 ψ3,0,0= ψ3,0,0(0) + 27 √ 5 35aµZ P n0 >3 1 1− 9 n02 +∞ R 0 Rn0 ,2(r)R3,0(r) r drψ (0) n0 ,2,0 = ψ(0)3,0,0−µ2 Z2 a 792√3 588245ψ (0) 4,2,0+45 √ 42 229376ψ (0) 5,2,0+ 656 √ 7 2066715ψ (0) 6,2,0+ 1809 √ 7 7812500ψ (0) 7,2,0+ · · · n = 3 l = 1 m = 0 ψ3,1,0= ψ3,1,0(0) +5aµZ18 P n0 >1 n0 6=3 1 1− 9 n02 +∞ R 0 Rn0 ,1(r)R3,1(r) r drψ (0) n0 ,1,0+ 27√21 35aµZ P n0 >3 1 1− 9 n02 +∞ R 0 Rn0 ,3(r)R3,1(r) r drψ (0) n0 ,3,0 = ψ(0)3,1,0+µ2 Z2a −1923125ψ(0)2,1,0+60288√10 2941225 ψ (0) 4,1,0+231 √ 5 16384ψ (0) 5,1,0+35648 √ 35 10333575ψ (0) 6,1,0+310191 √ 14 78125000 ψ (0) 7,1,0+ · · · −µ2 Z2a 10368 √ 10 20588575ψ (0) 4,3,0+ 81√5 458752ψ (0) 5,3,0+ 128√10 3444525ψ (0) 6,3,0+ 567√3 39062500ψ (0) 7,3,0+ · · · n = 3 l = 1 m = ±1 ψ3,1,±1= ψ3,1,±1(0) −5aµZ9 P n0 >3 1 1− 9 n02 +∞ R 0 Rn0 ,1(r)R3,1(r) r drψ (0) n0 ,1,±1+ 27√14 35aµZ P n0 >3 1 1− 9 n02 +∞ R 0 Rn0 ,3(r)R3,1(r) r drψ (0) n0 ,3,±1 = ψ(0)3,1,±1+µ2 Z2a 96 3125ψ (0) 2,1,±1− 30144√10 2941225 ψ (0) 4,1,±1− 231√5 32768ψ (0) 5,1,±1− 17824√35 10333575ψ (0) 6,1,±1− 310191√14 156250000ψ (0) 7,1,±1+ · · · −µ2 Z2a 6912 √ 15 20588575ψ (0) 4,3,±1+ 27√30 458752ψ (0) 5,3,±1+ 256√15 10333575ψ (0) 6,3,±1+ 567√2 39062500ψ (0) 7,3,±1+ · · · n = 3 l = 2 m = 0 ψ3,2,0= ψ3,2,0(0) + 9√5 5aµZ P n0 >0 n0 6=3 1 1− 9 n02 +∞ R 0 Rn0 ,0(r)R3,2(r) r drψ (0) n0 ,0,0+ 18 7aµZ P n0 >3 1 1− 9 n02 +∞ R 0 Rn0 ,2(r)R3,2(r) r drψ (0) n0 ,2,0 +54 √ 5 35aµZ P n0 >4 1 1− 9 n02 +∞ R 0 Rn0 ,4(r)R3,2(r) r drψ (0) n0 ,4,0 = ψ(0)3,2,0+µ2 Z2a − √ 6 480ψ (0) 1,0,0+32 √ 3 9375ψ (0) 2,0,0+ 128 √ 6 1764735ψ (0) 4,0,0+ 5 √ 30 196608ψ (0) 5,0,0+29524532 ψ (0) 6,0,0+ 392 √ 42 29296875ψ (0) 7,0,0+ · · · +µ2 Z2a 13824√6 4117715ψ (0) 4,2,0+675 √ 21 802816ψ (0) 5,2,0+ 9472 √ 14 14467005ψ (0) 6,2,0+ 7371 √ 14 15625000ψ (0) 7,2,0+ · · · +µ2 Z2a 135√105 1605632ψ (0) 5,4,0+ 512 √ 42 4822335ψ (0) 6,4,0+ 7749 √ 231 214843750ψ (0) 7,4,0+ · · · n = 3 l = 2 m = ±1 ψ3,2,±1= ψ3,2,±1(0) +7aµZ9 P n0 >3 1 1− 9 n02 +∞ R 0 Rn0 ,2(r)R3,2(r) r drψ (0) n0 ,2,±1+ 9√6 7aµZ P n0 >4 1 1− 9 n02 +∞ R 0 Rn0 ,4(r)R3,2(r) r drψ (0) n0 ,4,±1 = ψ(0)3,2,±1+µ2 Z2a 6912√6 4117715ψ (0) 4,2,±1+675 √ 21 1605632ψ (0) 5,2,±1+4736 √ 14 14467005ψ (0) 6,2,±1+7371 √ 14 31250000ψ (0) 7,2,±1+ · · · +µ2 Z2a 675 √ 14 3211264ψ (0) 5,4,±1+ 512√35 4822335ψ (0) 6,4,±1+ 7749√770 429687500ψ (0) 7,4,±1+ · · · n = 3 l = 2 m = ±2 ψ3,2,±2= ψ3,2,±2(0) − 18 √ 5 35aµZ P n0 >3 1 1− 9 n02 +∞ R 0 Rn0 ,2(r)R3,2(r) r drψ (0) n0 ,2,±2+ 9√3 7aµZ P n0 >4 1 1− 9 n02 +∞ R 0 Rn0 ,4(r)R3,2(r) r drψ (0) n0 ,4,±2 = ψ(0)3,2,±2−µ2 Z2a 13824√6 4117715ψ (0) 4,2,±2+675 √ 21 802816ψ (0) 5,2,±2+ 9472 √ 14 14467005ψ (0) 6,2,±2+ 7371 √ 14 15625000ψ (0) 7,2,±2+ · · · +µ2 Z2a 675 √ 7 3211264ψ (0) 5,4,±2+ 256√70 4822335ψ (0) 6,4,±2+ 7749√385 429687500ψ (0) 7,4,±2+ · · ·
4
Application to quantum chemical calculations
In the previous section, we have obtained first order perturbative corrections to the eigen-functions of a smeared Coulomb potential along an arbitrary axis. Such a potential can represent the average potential felt by an electron bounded to a vibrating nucleus of ef-fective charge Z, efef-fective reduced mass µ and efef-fective classical vibrating amplitude equal to
q
2nvib+1
2a (in harmonic quantum level nvib). Hence, it is hoped that these approximate
eigenstates could be appropriate to describe the electron density of the effective electronic Hamiltonians solved in the EN-MFCI method11.
Although the model Hamiltonian depends upon 3 parameters, µ, Z, and a, its first order approximate eigenstates are parametrized by only two independent ones. They can conveniently be chosen as, c = µZ and b = µZa . Furthermore, since the product b × c factorizes the first order perturbative term in the approximate DHOs expansion, Eq.(18), we propose to just add the functions made of these corrective terms to standard basis sets, and leave it to molecular orbital calculations to optimize their linear combination with the other orbitals.
More precisely, instead of using ψn,l,m(b, c) of Eq.(18) (making explicit the dependency
upon the free parameters), we propose to use ψn,l,m(1) (c) b · c = (−1) m16n2 r l + 1 2 X (n0,l0)6=(n,l) n02 (n02−n2) q l0+1 2 l0 l 2 0 0 0 l0 l 2 −m m 0 +∞ R 0 Rn0 ,l0(x)Rn,l(x) x dxψ (0) n0,l0,m(c), (19)
which only depends upon c through the ψ(0)n0,l0,m(c) functions. Setting DHO
(1)
n,l,m(c) =
ψn,l,m(1) (c) b·c ,
its linear combination coefficients in the occupied molecular orbitals, obtained for example in a HF calculation, implicitly determine the corresponding coefficient products, b · c, and consequently, if c is given, the free parameters b. For instance, in H2, there is only one
occupied HF orbital of the form, ψ1 =
P i uiχi + P j vjDHO (1)
j where the χi’s are standard
quantum numbers, indexed by j. The ui, vj’s are scalar coefficients. One can avoid the
explicit optimization of the bj · cj values for each DHOj(cj), and just use the vj’s values.
An important issue is to calculate efficiently multicenter integrals involving the χi’s and
DHOj(1)’s orbitals, the former being usually GTOs and the latter ETOs. We will adopt a strategy similar to that of the SMILES module14, optionally included in the MOLPRO
quantum chemistry package15to deal with STOs. That is to say, the DHO(1)’s will be trans-formed into an GTO-expansion, as also in Ref.16. However, different Cartesian prefactors will be mixed in the GTO-expansion of a given DHO. The BDF code17–19 has been modi-fied to accept contracted GTO having such mixed Cartesian prefactors. Full computational details will be provided in the second paper of this series20. Note that, a recently proposed,
alternative strategy could be considered in the future21.
A hyperbolic cosine factor can be associated to ETO to provide a “double zeta” character to a minimal basis set22. This can be considered for DHO, as well. However, it seems more natural within our framework to combine different sets of DHO corresponding to atoms in different ionization states, to obtain multi-zeta basis sets. This last option preserves the desirable property for the DHOs to have associated energy eigenvalues, which can be used for truncation purposes.
Beside their application to the EN-MFCI method, which will be investigated in the next paper of the series, DHOs could also be useful for “clamped nuclei” quantum chemistry cal-culations, as they can be regarded as naturally “sp-hybridized” (unlike spherical harmonics STO). Actually, one could develop other model Hamiltonian distorted from spherical sym-metry in more than one direction and obtain their first order approximate eigenfunctions by following the approach presented in this paper. For example, generalization of our con-voluted Coulomb potentials could provide model Hamitonians leading to naturally sp2- or sp3-hybridized DHOs.
Another possible interest for traditional quantum chemistry to be assessed in the future, is the ability of DHOs to replace advantageously polarization orbital sets, since high angular
momentum atomic primitives are already contracted within DHOs.
Acknowledgment
The authors were partially supported by ERC grant SCAPDE, 320845, and by the D¨oblin f´ed´eration de Recherches of CNRS.
References
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(14) J Fern´andez Rico, Ignacio Ema, Rafael Lopez, Guillermo Ram´ırez, Kazuhiro Ishida, ??5-SMILES: A package for molecular calculations with STOsoftware, third gener-ation1452008 (Telhat Ozdogan and Maria Belen Ruiz, editors, Transworld Research Network)
(15) Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Sch¨utz, M.; Celani, P.; Gy¨orffy, W.; Kats, D.; Korona, T.; Lindh, R.; Mitrushenkov, A.; Rauhut, G.; Shama-sundar, K. R.; Adler, T. B.; Amos, R. D.; Bennie, S. J.; Bernhardsson, A.; Berning, A.; Cooper, D. L.; Deegan, M. J. O.; Dobbyn, A. J.; Eckert, F.; Goll, E.; Hampel, C.; Hesselmann, A.; Hetzer, G.; Hrenar, T.; Jansen, G.; K¨oppl, C.; Lee, S. J. R.; Liu, Y.; Lloyd, A. W.; Ma, Q.; Mata, R. A.; May, A. J.; McNicholas, S. J.; Meyer, W.; III, T. F. M.; Mura, M. E.; Nicklass, A.; O’Neill, D. P.; Palmieri, P.; Peng, D.; Pfl¨uger, K.; Pitzer, R.; Reiher, M.; Shiozaki, T.; Stoll, H.; Stone, A. J.; Tarroni, R.; Thorsteins-son, T.; Wang, M.; Welborn, M. MOLPRO, version 2018.2, a package of ab initio programs. 2018; see http://www.molpro.net
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(20) P. Cassam-Chena¨ı, B. Suo, “Smeared Coulomb potential orbitals: II- Application to EN-MFCI calculations”, to be published
(21) M. Caffarel, ??1510641012019
b = 32;|ψ(~r)|2= 0.00004 b = 32;|ψ(~r)|2= 0.00016 b = 32;|ψ(~r)|2= 0.00032 b = 32;|ψ(~r)|2= 0.00064
b = 1;|ψ(~r)|2= 0.00004 b = 1;|ψ(~r)|2= 0.00016 b = 1;|ψ(~r)|2= 0.00032 b = 1;|ψ(~r)|2= 0.00064
Figure 1: Comparison of hydrogen smeared Coulomb potential 1s orbital squared density levels.
b = 32;|ψ(~r)|2= 0.00004 b = 32;|ψ(~r)|2= 0.00016 b = 32;|ψ(~r)|2= 0.00032 b = 32;|ψ(~r)|2= 0.00064
b = 1;|ψ(~r)|2= 0.00004 b = 1;|ψ(~r)|2= 0.00016 b = 1;|ψ(~r)|2= 0.00032 b = 1;|ψ(~r)|2= 0.00064
Figure 2: Comparison of hydrogen smeared Coulomb potential 2s orbital squared density levels.
b = 32;|ψ(~r)|2= 0.00004 b = 32;|ψ(~r)|2= 0.00016 b = 32;|ψ(~r)|2= 0.00032 b = 32;|ψ(~r)|2= 0.00064
b = 1;|ψ(~r)|2= 0.00004 b = 1;|ψ(~r)|2= 0.00016 b = 1;|ψ(~r)|2= 0.00032 b = 1;|ψ(~r)|2= 0.00064
Figure 3: Comparison of hydrogen smeared Coulomb potential 2pz orbital squared density
b = 32;|ψ(~r)|2= 0.00002 b = 32;|ψ(~r)|2= 0.00004 b = 32;|ψ(~r)|2= 0.00008 b = 32;|ψ(~r)|2= 0.00016
b = 1;|ψ(~r)|2= 0.00002 b = 1;|ψ(~r)|2= 0.00004 b = 1;|ψ(~r)|2= 0.00008 b = 1;|ψ(~r)|2= 0.00016
Figure 4: Comparison of hydrogen smeared Coulomb potential 2p±1 (m = ±1) orbital
b = 32;|ψ(~r)|2= 0.00004 b = 32;|ψ(~r)|2= 0.00016 b = 32;|ψ(~r)|2= 0.00032 b = 32;|ψ(~r)|2= 0.00064
b = 1;|ψ(~r)|2= 0.00004 b = 1;|ψ(~r)|2= 0.00016 b = 1;|ψ(~r)|2= 0.00032 b = 1;|ψ(~r)|2= 0.00064
Figure 5: Comparison of hydrogen smeared Coulomb potential 3s orbital squared density levels.
b = 32;|ψ(~r)|2= 0.00004 b = 32;|ψ(~r)|2= 0.00008 b = 32;|ψ(~r)|2= 0.00016 b = 32;|ψ(~r)|2= 0.00032
b = 1;|ψ(~r)|2= 0.00004 b = 1;|ψ(~r)|2= 0.00008 b = 1;|ψ(~r)|2= 0.00016 b = 1;|ψ(~r)|2= 0.00032
Figure 6: Comparison of hydrogen smeared Coulomb potential 3pz orbital squared density
b = 32;|ψ(~r)|2= 0.00008 b = 32;|ψ(~r)|2= 0.00016 b = 32;|ψ(~r)|2= 0.00032 b = 32;|ψ(~r)|2= 0.00064
b = 1;|ψ(~r)|2= 0.00008 b = 1;|ψ(~r)|2= 0.00016 b = 1;|ψ(~r)|2= 0.00032 b = 1;|ψ(~r)|2= 0.00064
Figure 7: Comparison of hydrogen smeared Coulomb potential 3p±1 (m = ±1) orbital
b = 32;|ψ(~r)|2= 0.00004 b = 32;|ψ(~r)|2= 0.00016 b = 32;|ψ(~r)|2= 0.00032 b = 32;|ψ(~r)|2= 0.00064
b = 1;|ψ(~r)|2= 0.00004 b = 1;|ψ(~r)|2= 0.00016 b = 1;|ψ(~r)|2= 0.00032 b = 1;|ψ(~r)|2= 0.00064
Figure 8: Comparison of hydrogen smeared Coulomb potential 3dz2 orbital squared density levels.
b = 32;|ψ(~r)|2= 0.00004 b = 32;|ψ(~r)|2= 0.00008 b = 32;|ψ(~r)|2= 0.00016 b = 32;|ψ(~r)|2= 0.00032
b = 1;|ψ(~r)|2= 0.00004 b = 1;|ψ(~r)|2= 0.00008 b = 1;|ψ(~r)|2= 0.00016 b = 1;|ψ(~r)|2= 0.00032
Figure 9: Comparison of hydrogen smeared Coulomb potential 3d±1 (m = ±1) orbital
b = 32;|ψ(~r)|2= 0.00002 b = 32;|ψ(~r)|2= 0.00004 b = 32;|ψ(~r)|2= 0.00008 b = 32;|ψ(~r)|2= 0.00016
b = 1;|ψ(~r)|2= 0.00002 b = 1;|ψ(~r)|2= 0.00004 b = 1;|ψ(~r)|2= 0.00008 b = 1;|ψ(~r)|2= 0.00016
Figure 10: Comparison of hydrogen smeared Coulomb potential 3d±2 (m = ±2) orbital
App
endix:
First
order
corrected
eigenstates
up
to
n=7
n = 1 E 0 1 = − µZ 2 2 l = 0 E 1 1,0 ,0 = 0 n = 2 E 0 2 = − µZ 2 8 l = 0 E 1 2,0 ,0 = 0 l = 1 E 1 2,1 ,0 = − µ 3Z 4 120 a E 1 2,1 ,± 1 = µ 3Z 4 240 a n = 3 E 0 3 = − µZ 2 18 l = 0 E 1 3,0 ,0 = 0 l = 1 E 1 3,1 ,0 = − µ 3Z 4 405 a E 1 3,1 ,± 1 = µ 3Z 4 810 a l = 2 E 1 3,2 ,0 = − µ 3Z 4 2835 a E 1 3,2 ,± 1 = − µ 3Z 4 5670 a E 1 3,2 ,± 2 = µ 3Z 4 2835 a n = 4 E 0 4 = − µZ 2 32 l = 0 E 1 4,0 ,0 = 0 l = 1 E 1 4,1 ,0 = − µ 3Z 4 960 a E 1 4,1 ,± 1 = µ 3Z 4 1920 a l = 2 E 1 4,2 ,0 = − µ 3Z 4 6720 a E 1 4,2 ,± 1 = − µ 3Z 4 13440 a E 1 4,2 ,± 2 = µ 3Z 4 6720 a l = 3 E 1 4,3 ,0 = − µ 3Z 4 20160 a E 1 4,3 ,± 1 = − µ 3Z 4 26880 a E 1 4,3 ,± 2 = 0 E 1 4,3 ,± 3 = µ 3Z 4 16128 an = 5 E 0 5= − µZ 2 50 l = 0 E 1 5,0 ,0 = 0 l = 1 E 1 5,1 ,0 = − µ 3Z 4 1875 a E 1 5,1 ,± 1 = µ 3Z 4 3750 a l = 2 E 1 5,2 ,0 = − µ 3Z 4 13125 a E 1 5,2 ,± 1 = − µ 3Z 4 26250 a E 1 5,2 ,± 2 = µ 3Z 4 13125 a l = 3 E 1 5,3 ,0 = − µ 3Z 4 39375 a E 1 5,3 ,± 1 = − µ 3Z 4 52500 a E 1 5,3 ,± 2 = 0 E 1 5,3 ,± 3 = µ 3Z 4 31500 a l = 4 E 1 5,4 ,0 = − µ 3Z 4 86625 a E 1 5,4 ,± 1 = − 17 µ 3Z 4 1732500 a E 1 5,4 ,± 2 = − 2 µ 3Z 4 433125 a E 1 5,4 ,± 3 = µ 3Z 4 247500 a E 1 5,4 ,± 4 = µ 3Z 4 61875 a n = 6 E 0 6 = − µZ 2 72 l = 0 E 1 6,0 ,0 = 0 l = 1 E 1 6,1 ,0 = − µ 3Z 4 3240 a E 1 6,1 ,± 1 = µ 3Z 4 6480 a l = 2 E 1 6,2 ,0 = − µ 3Z 4 22680 a E 1 6,2 ,± 1 = − µ 3Z 4 45360 a E 1 6,2 ,± 2 = µ 3Z 4 22680 a l = 3 E 1 6,3 ,0 = − µ 3Z 4 68040 a E 1 6,3 ,± 1 = − µ 3Z 4 90720 a E 1 6,3 ,± 2 = 0 E 1 6,3 ,± 3 = µ 3Z 4 54432 a l = 4 E 1 6,4 ,0 = − µ 3Z 4 149688 a E 1 6,4 ,± 1 = − 17 µ 3Z 4 2993760 a E 1 6,4 ,± 2 = − µ 3Z 4 374220 a E 1 6,4 ,± 3 = µ 3Z 4 427680 a E 1 6,4 ,± 4 = µ 3Z 4 106920 a l = 5 E 1 6,5 ,0 = − µ 3Z 4 277992 a E 1 6,5 ,± 1 = − µ 3Z 4 308880 a E 1 6,5 ,± 2 = − µ 3Z 4 463320 a E 1 6,5 ,± 3 = − µ 3Z 4 2779920 a E 1 6,5 ,± 4 = µ 3Z 4 463320 a E 1 6,5 ,± 5 = µ 3Z 4 185328 a n = 7 E 0 7 = − µZ 2 98 l = 0 E 1 7,0 ,0 = 0 l = 1 E 1 7,1 ,0 = − µ 3Z 4 5145 a E 1 7,1 ,± 1 = µ 3Z 4 10290 a l = 2 E 1 7,2 ,0 = − µ 3Z 4 36015 a E 1 7,2 ,± 1 = − µ 3Z 4 72030 a E 1 7,2 ,± 2 = µ 3Z 4 36015 a l = 3 E 1 7,3 ,0 = − µ 3Z 4 108045 a E 1 7,3 ,± 1 = − µ 3Z 4 144060 a E 1 7,3 ,± 2 = 0 E 1 7,3 ,± 3 = µ 3Z 4 86436 a l = 4 E 1 7,4 ,0 = − µ 3Z 4 237699 a E 1 7,4 ,± 1 = − 17 µ 3Z 4 4753980 a E 1 7,4 ,± 2 = − 2 µ 3Z 4 1188495 a E 1 7,4 ,± 3 = µ 3Z 4 679140 a E 1 7,4 ,± 4 = µ 3Z 4 169785 a l = 5 E 1 7,5 ,0 = − µ 3Z 4 441441 a E 1 7,5 ,± 1 = − µ 3Z 4 490490 a E 1 7,5 ,± 2 = − µ 3Z 4 735735 a E 1 7,5 ,± 3 = − µ 3Z 4 4414410 a E 1 7,5 ,± 4 = µ 3Z 4 735735 a E 1 7,5 ,± 5 = µ 3Z 4 294294 a l = 6 E 1 7,6 ,0 = − µ 3Z 4 735735 a E 1 7,6 ,± 1 = − µ 3Z 4 792330 a E 1 7,6 ,± 2 = − µ 3Z 4 1030029 a E 1 7,6 ,± 3 = − µ 3Z 4 2060058 a E 1 7,6 ,± 4 = µ 3Z 4 5150145 a E 1 7,6 ,± 5 = µ 3Z 4 936390 a E 1 7,6 ,± 6 = µ 3Z 4 468195 a T able 4: Zero order eigen v alues and first order corrections up to n=7
n = 1 < 100 |320 > = √ 6 480 < 100 |420 > = 104 28125 < 100 |520 > = 5 √ 14 6804 < 100 |620 > = 5744 √ 21 12353145 < 100 |720 > = 3299 √ 21 8847360 n = 2 < 200 |320 > = − 32 √ 3 9375 < 200 |420 > = − 2 √ 2 1215 < 200 |520 > = − 3680 √ 7 7411887 < 200 |620 > = − 29 √ 42 215040 < 200 |720 > = − 63584 √ 42 645700815 < 210 |310 > = 192 3125 < 210 |410 > = 56 √ 10 6075 < 210 |510 > = 8768 √ 5 1058841 < 210 |610 > = 201 √ 35 89600 < 210 |710 > = 2926784 √ 14 1076168025 < 210 |430 > = 8 √ 10 14175 < 210 |530 > = 512 √ 5 823543 < 210 |630 > = 123 √ 10 358400 < 210 |730 > = 541184 √ 3 1076168025 < 211 |311 > = − 96 3125 < 211 |411 > = − 28 √ 10 6075 < 211 |511 > = − 4384 √ 5 1058841 < 211 |611 > = − 201 √ 35 179200 < 211 |711 > = − 1463392 √ 14 1076168025 < 211 |431 > = 16 √ 15 42525 < 211 |531 > = 512 √ 30 2470629 < 211 |631 > = 41 √ 15 179200 < 211 |731 > = 541184 √ 2 1076168025 n = 3 < 300 |420 > = − 792 √ 3 588245 < 300 |520 > = − 45 √ 42 229376 < 300 |620 > = − 656 √ 7 2066715 < 300 |720 > = − 1809 √ 7 7812500 < 310 |210 > = − 192 3125 < 310 |410 > = 60288 √ 10 2941225 < 310 |510 > = 231 √ 5 16384 < 310 |610 > = 35648 √ 35 10333575 < 310 |710 > = 310191 √ 14 78125000 < 310 |430 > = − 10368 √ 10 20588575 < 310 |530 > = − 81 √ 5 458752 < 310 |630 > = − 128 √ 10 3444525 < 310 |730 > = − 567 √ 3 39062500 < 311 |211 > = 96 3125 < 311 |411 > = − 30144 √ 10 2941225 < 311 |511 > = − 231 √ 5 32768 < 311 |611 > = − 17824 √ 35 10333575 < 311 |711 > = − 310191 √ 14 156250000 < 311 |431 > = − 6912 √ 15 20588575 < 311 |531 > = − 27 √ 30 458752 < 311 |631 > = − 256 √ 15 10333575 < 311 |731 > = − 567 √ 2 39062500 < 320 |100 > = − √ 6 480 < 320 |200 > = 32 √ 3 9375 < 320 |400 > = 128 √ 6 1764735 < 320 |500 > = 5 √ 30 196608 < 320 |600 > = 32 295245 < 320 |700 > = 392 √ 42 29296875 < 320 |420 > = 13824 √ 6 4117715 < 320 |520 > = 675 √ 21 802816 < 320 |620 > = 9472 √ 14 14467005 < 320 |720 > = 7371 √ 14 15625000 < 320 |540 > = 135 √ 105 1605632 < 320 |640 > = 512 √ 42 4822335 < 320 |740 > = 7749 √ 231 214843750 < 321 |421 > = 6912 √ 6 4117715 < 321 |521 > = 675 √ 21 1605632 < 321 |621 > = 4736 √ 14 14467005 < 321 |721 > = 7371 √ 14 31250000 < 321 |541 > = 675 √ 14 3211264 < 321 |641 > = 512 √ 35 4822335 < 321 |741 > = 7749 √ 770 429687500 < 322 |422 > = − 13824 √ 6 4117715 < 322 |522 > = − 675 √ 21 802816 < 322 |622 > = − 9472 √ 14 14467005 < 322 |722 > = − 7371 √ 14 15625000 < 322 |542 > = 675 √ 7 3211264 < 322 |642 > = 256 √ 70 4822335 < 322 |742 > = 7749 √ 385 429687500
n = 4 < 400 |320 > = − 128 √ 6 1764735 < 400 |520 > = − 91520 √ 14 301327047 < 400 |620 > = − 6368 √ 21 41015625 < 400 |720 > = − 388039424 √ 21 3501552321135 < 410 |210 > = − 56 √ 10 6075 < 410 |310 > = − 60288 √ 10 2941225 < 410 |510 > = 2001280 √ 2 43046721 < 410 |610 > = 590424 √ 14 68359375 < 410 |710 > = 20835775744 √ 35 5835920535225 < 410 |530 > = − 20480 √ 2 33480783 < 410 |630 > = − 27264 68359375 < 410 |730 > = − 81342464 √ 30 1945306845075 < 411 |211 > = 28 √ 10 6075 < 411 |311 > = 30144 √ 10 2941225 < 411 |511 > = − 1000640 √ 2 43046721 < 411 |611 > = − 295212 √ 14 68359375 < 411 |711 > = − 10417887872 √ 35 5835920535225 < 411 |531 > = − 40960 √ 3 100442349 < 411 |631 > = − 9088 √ 6 68359375 < 411 |731 > = − 162684928 √ 5 1945306845075 < 420 |100 > = − 104 28125 < 420 |200 > = 2 √ 2 1215 < 420 |300 > = 792 √ 3 588245 < 420 |500 > = 2360 √ 5 43046721 < 420 |600 > = 414 √ 6 9765625 < 420 |700 > = 4159512 √ 7 129687123005 < 420 |320 > = − 13824 √ 6 4117715 < 420 |520 > = 5081600 √ 14 2109289329 < 420 |620 > = 91008 √ 21 95703125 < 420 |720 > = 2156342272 √ 21 3501552321135 < 420 |540 > = − 163840 √ 70 2109289329 < 420 |640 > = − 9216 √ 7 478515625 < 420 |740 > = 23166976 √ 154 4279675059165 < 421 |321 > = − 6912 √ 6 4117715 < 421 |521 > = 2540800 √ 14 2109289329 < 421 |621 > = 45504 √ 21 95703125 < 421 |721 > = 1078171136 √ 21 3501552321135 < 421 |541 > = − 819200 √ 21 6327867987 < 421 |641 > = − 1536 √ 210 478515625 < 421 |741 > = 23166976 √ 1155 12839025177495 < 422 |322 > = 13824 √ 6 4117715 < 422 |522 > = − 5081600 √ 14 2109289329 < 422 |622 > = − 91008 √ 21 95703125 < 422 |722 > = − 2156342272 √ 21 3501552321135 < 422 |542 > = − 409600 √ 42 6327867987 < 422 |642 > = − 1536 √ 105 478515625 < 422 |742 > = 11583488 √ 2310 12839025177495 < 430 |210 > = − 8 √ 10 14175 < 430 |310 > = 10368 √ 10 20588575 < 430 |510 > = 3200 √ 2 100442349 < 430 |610 > = 648 √ 14 68359375 < 430 |710 > = 3048192 √ 35 648435615025 < 430 |530 > = 1638400 √ 2 903981141 < 430 |630 > = 79872 68359375 < 430 |730 > = 7035879424 √ 30 52523284817025 < 430 |650 > = 12288 √ 35 150390625 < 430 |750 > = 873463808 √ 105 23110245319491 < 431 |211 > = − 16 √ 15 42525 < 431 |311 > = 6912 √ 15 20588575 < 431 |511 > = 6400 √ 3 301327047 < 431 |611 > = 432 √ 21 68359375 < 431 |711 > = 1016064 √ 210 648435615025 < 431 |531 > = 409600 √ 2 301327047 < 431 |631 > = 59904 68359375 < 431 |731 > = 1758969856 √ 30 17507761605675 < 431 |651 > = 18432 √ 14 150390625 < 431 |751 > = 436731904 √ 42 7703415106497 < 432 |652 > = 18432 √ 5 107421875 < 432 |752 > = 3057123328 √ 15 38517075532485 < 433 |533 > = − 2048000 √ 2 903981141 < 433 |633 > = − 19968 13671875 < 433 |733 > = − 1758969856 √ 30 10504656963405 < 433 |653 > = 12288 √ 5 107421875 < 433 |753 > = 6114246656 √ 15 115551226597455
n = 5 < 500 |320 > = − 5 √ 30 196608 < 500 |420 > = − 2360 √ 5 43046721 < 500 |620 > = − 33826640 √ 105 544685916621 < 500 |720 > = − 91685 √ 105 2176782336 < 510 |210 > = − 8768 √ 5 1058841 < 510 |310 > = − 231 √ 5 16384 < 510 |410 > = − 2001280 √ 2 43046721 < 510 |610 > = 4537728960 √ 7 181561972207 < 510 |710 > = 708547 √ 70 181398528 < 510 |430 > = − 3200 √ 2 100442349 < 510 |630 > = − 64694400 √ 2 181561972207 < 510 |730 > = − 2975 √ 15 40310784 < 511 |211 > = 4384 √ 5 1058841 < 511 |311 > = 231 √ 5 32768 < 511 |411 > = 1000640 √ 2 43046721 < 511 |611 > = − 2268864480 √ 7 181561972207 < 511 |711 > = − 708547 √ 70 362797056 < 511 |431 > = − 6400 √ 3 301327047 < 511 |631 > = − 43129600 √ 3 181561972207 < 511 |731 > = − 2975 √ 10 40310784 < 520 |100 > = − 5 √ 14 6804 < 520 |200 > = 3680 √ 7 7411887 < 520 |300 > = 45 √ 42 229376 < 520 |400 > = 91520 √ 14 301327047 < 520 |600 > = 3258720 √ 21 181561972207 < 520 |700 > = 18865 √ 2 362797056 < 520 |320 > = − 675 √ 21 802816 < 520 |420 > = − 5081600 √ 14 2109289329 < 520 |620 > = 98092800 √ 6 25937424601 < 520 |720 > = 1008175 √ 6 544195584 < 520 |640 > = − 54720000 √ 2 181561972207 < 520 |740 > = − 30625 √ 11 665127936 < 521 |321 > = − 675 √ 21 1605632 < 521 |421 > = − 2540800 √ 14 2109289329 < 521 |621 > = 49046400 √ 6 25937424601 < 521 |721 > = 1008175 √ 6 1088391168 < 521 |641 > = − 18240000 √ 15 181561972207 < 521 |741 > = − 30625 √ 330 3990767616 < 522 |322 > = 675 √ 21 802816 < 522 |422 > = 5081600 √ 14 2109289329 < 522 |622 > = − 98092800 √ 6 25937424601 < 522 |722 > = − 1008175 √ 6 544195584 < 522 |642 > = − 9120000 √ 30 181561972207 < 522 |742 > = − 30625 √ 165 3990767616 < 530 |210 > = − 512 √ 5 823543 < 530 |310 > = 81 √ 5 458752 < 530 |410 > = 20480 √ 2 33480783 < 530 |610 > = 2695680 √ 7 181561972207 < 530 |710 > = 637 √ 70 161243136 < 530 |430 > = − 1638400 √ 2 903981141 < 530 |630 > = 372940800 √ 2 181561972207 < 530 |730 > = 145775 √ 15 408146688 < 530 |650 > = − 76800000 √ 70 1997181694277 < 530 |750 > = 153125 √ 210 71833817088 < 531 |211 > = − 512 √ 30 2470629 < 531 |311 > = 27 √ 30 458752 < 531 |411 > = 40960 √ 3 100442349 < 531 |611 > = 898560 √ 42 181561972207 < 531 |711 > = 637 √ 105 241864704 < 531 |431 > = − 409600 √ 2 301327047 < 531 |631 > = 279705600 √ 2 181561972207 < 531 |731 > = 145775 √ 15 544195584 < 531 |651 > = − 230400000 √ 7 1997181694277 < 531 |751 > = 153125 √ 21 23944605696 < 532 |652 > = − 23040000 √ 10 285311670611 < 532 |752 > = 214375 √ 30 47889211392 < 533 |433 > = 2048000 √ 2 903981141 < 533 |633 > = − 466176000 √ 2 181561972207 < 533 |733 > = − 728875 √ 15 1632586752 < 533 |653 > = − 15360000 √ 10 285311670611 < 533 |753 > = 214375 √ 30 71833817088 < 540 |320 > = − 135 √ 105 1605632 < 540 |420 > = 163840 √ 70 2109289329 < 540 |620 > = 552960 √ 30 181561972207 < 540 |720 > = 1715 √ 30 725594112 < 540 |640 > = 691200000 √ 10 1997181694277 < 540 |740 > = 1071875 √ 55 16461916416 < 540 |760 > = 37515625 √ 330 2282719076352 < 541 |321 > = − 675 √ 14 3211264 < 541 |421 > = 819200 √ 21 6327867987 < 541 |621 > = 2764800 181561972207 < 541 |721 > = 8575 725594112 < 541 |641 > = 587520000 √ 10 1997181694277 < 541 |741 > = 3644375 √ 55 65847665664 < 541 |761 > = 37515625 √ 77 1141359538176 < 542 |322 > = − 675 √ 7 3211264 < 542 |422 > = 409600 √ 42 6327867987 < 542 |622 > = 1382400 √ 2 181561972207 < 542 |722 > = 8575 √ 2 1451188224 < 542 |642 > = 276480000 √ 10 1997181694277 < 542 |742 > = 214375 √ 55 8230958208 < 542 |762 > = 7503125 √ 385 570679769088 < 543 |643 > = − 34560000 √ 10 285311670611 < 543 |743 > = − 1500625 √ 55 65847665664 < 543 |763 > = 7503125 √ 110 380453179392 < 544 |644 > = − 138240000 √ 10 285311670611 < 544 |744 > = − 1500625 √ 55 16461916416 < 544 |764 > = 37515625 √ 66 2282719076352
n = 6 < 600 |320 > = − 32 295245 < 600 |420 > = − 414 √ 6 9765625 < 600 |520 > = − 3258720 √ 21 181561972207 < 600 |720 > = − 12194800992 √ 14 11649042561240 5 < 610 |210 > = − 201 √ 35 89600 < 610 |310 > = − 35648 √ 35 10333575 < 610 |410 > = − 590424 √ 14 68359375 < 610 |510 > = − 4537728960 √ 7 181561972207 < 610 |710 > = 12213784164288 √ 10 582452128062025 < 610 |430 > = − 648 √ 14 68359375 < 610 |530 > = − 2695680 √ 7 181561972207 < 610 |730 > = − 17995074048 √ 105 582452128062025 < 611 |211 > = 201 √ 35 179200 < 611 |311 > = 17824 √ 35 10333575 < 611 |411 > = 295212 √ 14 68359375 < 611 |511 > = 2268864480 √ 7 181561972207 < 611 |711 > = − 6106892082144 √ 10 582452128062025 < 611 |431 > = − 432 √ 21 68359375 < 611 |531 > = − 898560 √ 42 181561972207 < 611 |731 > = − 17995074048 √ 70 582452128062025 < 620 |100 > = − 5744 √ 21 12353145 < 620 |200 > = 29 √ 42 215040 < 620 |300 > = 656 √ 7 2066715 < 620 |400 > = 6368 √ 21 41015625 < 620 |500 > = 33826640 √ 105 544685916621 < 620 |700 > = 11441201296 √ 3 34947127683721 5 < 620 |320 > = − 9472 √ 14 14467005 < 620 |420 > = − 91008 √ 21 95703125 < 620 |520 > = − 98092800 √ 6 25937424601 < 620 |720 > = 1091548749312 11649042561240 5 < 620 |540 > = − 552960 √ 30 181561972207 < 620 |740 > = − 44094468096 √ 66 128139468173645 5 < 621 |321 > = − 4736 √ 14 14467005 < 621 |421 > = − 45504 √ 21 95703125 < 621 |521 > = − 49046400 √ 6 25937424601 < 621 |721 > = 545774374656 116490425612405 < 621 |541 > = − 2764800 181561972207 < 621 |741 > = − 44094468096 √ 55 12813946817364 55 < 622 |322 > = 9472 √ 14 14467005 < 622 |422 > = 91008 √ 21 95703125 < 622 |522 > = 98092800 √ 6 25937424601 < 622 |722 > = − 1091548749312 11649042561240 5 < 622 |542 > = − 1382400 √ 2 181561972207 < 622 |742 > = − 22047234048 √ 110 12813946817364 55 < 630 |210 > = − 123 √ 10 358400 < 630 |310 > = 128 √ 10 3444525 < 630 |410 > = 27264 68359375 < 630 |510 > = 64694400 √ 2 181561972207 < 630 |710 > = 3040778496 √ 35 582452128062025 < 630 |430 > = − 79872 68359375 < 630 |530 > = − 372940800 √ 2 181561972207 < 630 |730 > = 322034786304 √ 30 58245212806202 5 < 630 |750 > = − 5982584832 √ 105 256278936347291 < 631 |211 > = − 41 √ 15 179200 < 631 |311 > = 256 √ 15 10333575 < 631 |411 > = 9088 √ 6 68359375 < 631 |511 > = 43129600 √ 3 181561972207 < 631 |711 > = 1013592832 √ 210 582452128062025 < 631 |431 > = − 59904 68359375 < 631 |531 > = − 279705600 √ 2 181561972207 < 631 |731 > = 241526089728 √ 30 58245212806202 5 < 631 |751 > = − 8973877248 √ 42 256278936347291 < 632 |752 > = − 62817140736 √ 15 1281394681736455 < 633 |433 > = 19968 13671875 < 633 |533 > = 466176000 √ 2 181561972207 < 633 |733 > = − 80508696576 √ 30 116490425612405 < 633 |753 > = − 41878093824 √ 15 1281394681736455 < 640 |320 > = − 512 √ 42 4822335 < 640 |420 > = 9216 √ 7 478515625 < 640 |520 > = 54720000 √ 2 181561972207 < 640 |720 > = 1134830592 √ 3 116490425612405 < 640 |540 > = − 691200000 √ 10 1997181694277 < 640 |740 > = 765716668416 √ 22 2819068299820201 < 640 |760 > = − 1147095613440 √ 33 36647887897662 613 < 641 |321 > = − 512 √ 35 4822335 < 641 |421 > = 1536 √ 210 478515625 < 641 |521 > = 18240000 √ 15 181561972207 < 641 |721 > = 567415296 √ 10 116490425612405 < 641 |541 > = − 587520000 √ 10 1997181694277 < 641 |741 > = 3254295840768 √ 22 14095341499101005 < 641 |761 > = − 229419122688 √ 770 36647887897662 613 < 642 |322 > = − 256 √ 70 4822335 < 642 |422 > = 1536 √ 105 478515625 < 642 |522 > = 9120000 √ 30 181561972207 < 642 |722 > = 567415296 √ 5 116490425612405 < 642 |542 > = − 276480000 √ 10 1997181694277 < 642 |742 > = 1531433336832 √ 22 14095341499101005 < 642 |762 > = − 458838245376 √ 154 36647887897662 613 < 643 |543 > = 34560000 √ 10 285311670611 < 643 |743 > = − 1340004169728 √ 22 14095341499101 005 < 643 |763 > = − 1376514736128 √ 11 36647887897662 613 < 644 |544 > = 138240000 √ 10 285311670611 < 644 |744 > = − 5360016678912 √ 22 14095341499101 005 < 644 |764 > = − 229419122688 √ 165 36647887897662 613 < 650 |430 > = − 12288 √ 35 150390625 < 650 |530 > = 76800000 √ 70 1997181694277 < 650 |730 > = 295034880 √ 42 256278936347291 < 650 |750 > = 1070622572544 √ 3 3331626172514783 < 651 |431 > = − 18432 √ 14 150390625 < 651 |531 > = 230400000 √ 7 1997181694277 < 651 |731 > = 177020928 √ 105 256278936347291 < 651 |751 > = 4817801576448 √ 3 16658130862573915 < 652 |432 > = − 18432 √ 5 107421875 < 652 |532 > = 23040000 √ 10 285311670611 < 652 |732 > = 619573248 √ 6 256278936347291 < 652 |752 > = 3211867717632 √ 3 16658130862573915
n = 6 c ontinue d ... < 653 |433 > = − 12288 √ 5 107421875 < 653 |533 > = 15360000 √ 10 285311670611 < 653 |733 > = 413048832 √ 6 256278936347291 < 653 |753 > = 535311286272 √ 3 16658130862573915 < 654 |754 > = − 3211867717632 √ 3 16658130862573915 < 655 |755 > = − 1605933858816 √ 3 3331626172514783
n = 7 < 700 |320 > = − 392 √ 42 29296875 < 700 |420 > = − 4159512 √ 7 129687123005 < 700 |520 > = − 18865 √ 2 362797056 < 700 |620 > = − 11441201296 √ 3 34947127683721 5 < 710 |210 > = − 2926784 √ 14 1076168025 < 710 |310 > = − 310191 √ 14 78125000 < 710 |410 > = − 20835775744 √ 35 5835920535225 < 710 |510 > = − 708547 √ 70 181398528 < 710 |610 > = − 12213784164288 √ 10 582452128062025 < 710 |430 > = − 3048192 √ 35 648435615025 < 710 |530 > = − 637 √ 70 161243136 < 710 |630 > = − 3040778496 √ 35 582452128062025 < 711 |211 > = 1463392 √ 14 1076168025 < 711 |311 > = 310191 √ 14 156250000 < 711 |411 > = 10417887872 √ 35 5835920535225 < 711 |511 > = 708547 √ 70 362797056 < 711 |611 > = 6106892082144 √ 10 582452128062025 < 711 |431 > = − 1016064 √ 210 648435615025 < 711 |531 > = − 637 √ 105 241864704 < 711 |631 > = − 1013592832 √ 210 582452128062025 < 720 |100 > = − 3299 √ 21 8847360 < 720 |200 > = 63584 √ 42 645700815 < 720 |300 > = 1809 √ 7 7812500 < 720 |400 > = 388039424 √ 21 3501552321135 < 720 |500 > = 91685 √ 105 2176782336 < 720 |600 > = 12194800992 √ 14 116490425612405 < 720 |320 > = − 7371 √ 14 15625000 < 720 |420 > = − 2156342272 √ 21 3501552321135 < 720 |520 > = − 1008175 √ 6 544195584 < 720 |620 > = − 1091548749312 116490425612405 < 720 |540 > = − 1715 √ 30 725594112 < 720 |640 > = − 1134830592 √ 3 116490425612405 < 721 |321 > = − 7371 √ 14 31250000 < 721 |421 > = − 1078171136 √ 21 3501552321135 < 721 |521 > = − 1008175 √ 6 1088391168 < 721 |621 > = − 545774374656 11649042561240 5 < 721 |541 > = − 8575 725594112 < 721 |641 > = − 567415296 √ 10 11649042561240 5 < 722 |322 > = 7371 √ 14 15625000 < 722 |422 > = 2156342272 √ 21 3501552321135 < 722 |522 > = 1008175 √ 6 544195584 < 722 |622 > = 1091548749312 116490425612405 < 722 |542 > = − 8575 √ 2 1451188224 < 722 |642 > = − 567415296 √ 5 11649042561240 5 < 730 |210 > = − 541184 √ 3 1076168025 < 730 |310 > = 567 √ 3 39062500 < 730 |410 > = 81342464 √ 30 1945306845075 < 730 |510 > = 2975 √ 15 40310784 < 730 |610 > = 17995074048 √ 105 58245212806202 5 < 730 |430 > = − 7035879424 √ 30 52523284817025 < 730 |530 > = − 145775 √ 15 408146688 < 730 |630 > = − 322034786304 √ 30 582452128062025 < 730 |650 > = − 295034880 √ 42 256278936347291 < 731 |211 > = − 541184 √ 2 1076168025 < 731 |311 > = 567 √ 2 39062500 < 731 |411 > = 162684928 √ 5 1945306845075 < 731 |511 > = 2975 √ 10 40310784 < 731 |611 > = 17995074048 √ 70 582452128062025 < 731 |431 > = − 1758969856 √ 30 17507761605675 < 731 |531 > = − 145775 √ 15 544195584 < 731 |631 > = − 241526089728 √ 30 582452128062025 < 731 |651 > = − 177020928 √ 105 256278936347291 < 732 |652 > = − 619573248 √ 6 256278936347291 < 733 |433 > = 1758969856 √ 30 10504656963405 < 733 |533 > = 728875 √ 15 1632586752 < 733 |633 > = 80508696576 √ 30 116490425612405 < 733 |653 > = − 413048832 √ 6 256278936347291 < 740 |320 > = − 7749 √ 231 214843750 < 740 |420 > = − 23166976 √ 154 4279675059165 < 740 |520 > = 30625 √ 11 665127936 < 740 |620 > = 44094468096 √ 66 1281394681736455 < 740 |540 > = − 1071875 √ 55 16461916416 < 740 |640 > = − 765716668416 √ 22 28190682998202 01 < 741 |321 > = − 7749 √ 770 429687500 < 741 |421 > = − 23166976 √ 1155 12839025177495 < 741 |521 > = 30625 √ 330 3990767616 < 741 |621 > = 44094468096 √ 55 1281394681736455 < 741 |541 > = − 3644375 √ 55 65847665664 < 741 |641 > = − 3254295840768 √ 22 14095341499101 005 < 742 |322 > = − 7749 √ 385 429687500 < 742 |422 > = − 11583488 √ 2310 12839025177495 < 742 |522 > = 30625 √ 165 3990767616 < 742 |622 > = 22047234048 √ 110 1281394681736455 < 742 |542 > = − 214375 √ 55 8230958208 < 742 |642 > = − 1531433336832 √ 22 14095341499101 005 < 743 |543 > = 1500625 √ 55 65847665664 < 743 |643 > = 1340004169728 √ 22 14095341499101005 < 744 |544 > = 1500625 √ 55 16461916416 < 744 |644 > = 5360016678912 √ 22 14095341499101005
n = 7 c ontinue d ... < 750 |430 > = − 873463808 √ 105 23110245319491 < 750 |530 > = − 153125 √ 210 71833817088 < 750 |630 > = 5982584832 √ 105 25627893634729 1 < 750 |650 > = − 1070622572544 √ 3 333162617251478 3 < 751 |431 > = − 436731904 √ 42 7703415106497 < 751 |531 > = − 153125 √ 21 23944605696 < 751 |631 > = 8973877248 √ 42 25627893634729 1 < 751 |651 > = − 4817801576448 √ 3 166581308625739 15 < 752 |432 > = − 3057123328 √ 15 38517075532485 < 752 |532 > = − 214375 √ 30 47889211392 < 752 |632 > = 62817140736 √ 15 12813946817364 55 < 752 |652 > = − 3211867717632 √ 3 166581308625739 15 < 753 |433 > = − 6114246656 √ 15 115551226597455 < 753 |533 > = − 214375 √ 30 71833817088 < 753 |633 > = 41878093824 √ 15 12813946817364 55 < 753 |653 > = − 535311286272 √ 3 166581308625739 15 < 754 |654 > = 3211867717632 √ 3 16658130862573915 < 755 |655 > = 1605933858816 √ 3 3331626172514783 < 760 |540 > = − 37515625 √ 330 2282719076352 < 760 |640 > = 1147095613440 √ 33 36647887897662613 < 761 |541 > = − 37515625 √ 77 1141359538176 < 761 |641 > = 229419122688 √ 770 36647887897662613 < 762 |542 > = − 7503125 √ 385 570679769088 < 762 |642 > = 458838245376 √ 154 36647887897662613 < 763 |543 > = − 7503125 √ 110 380453179392 < 763 |643 > = 1376514736128 √ 11 36647887897662613 < 764 |544 > = − 37515625 √ 66 2282719076352 < 764 |644 > = 229419122688 √ 165 36647887897662613 T able 5: Non-zero, first order coupling co efficien ts up to n=7 in µ 2Z 2 a units, that is to sa y, ψ (1) n,l ,m = P (n 0,l 0)6=( n,l ) < n 0 l 0 m |nl m > µ 2Z 2 a ψ (0) 0n,l 0,m . Only p ositiv e m -v alue s are tabulated, their negativ e coun te rparts giving the same coupling co efficien ts. Note that, all n um b ers b eing real, < n 0 l 0 m |nl m > = < nl m |n 0 l 0 m >
