Orthogonal Linear Combinations of Gaussian Type Orbitals

0.25 MB·English

by  Richard J. Mathar

Orthogonal Linear Combinations of Gaussian Type Orbitals
Richard J. Mathar
Leiden Observatory, P.O. Box 9513, 2300 RA Leiden, The Netherlands∗
(Dated: January 24, 2009)
The set of Gaussian Type Orbitals g(n ,n ,n ) of order (n+1)(n+2)/2 and of common n
1 2 3
≡
n +n +n 7, common center and exponential, is customized to define a set of 2n+1 linear
1 2 3
9 combinationst≤ ( n m n) such that each t dependson theazimuthaland polar angle of
n,m n,m
0 thesphericalcoordin−ate≤syste≤mliketherealorimaginarypartoftheassociatedSphericalHarmonic.
0 (Results cover both Hermite and Cartesian Gaussian Type Orbitals.) Overlap, kinetic energy and
2 Coulomb energy matrix elements are presented for generalized basis functions of the type rst
n,m
n (s=0,2,4...). In addition, normalization integrals g(n1,n2,n3)d3r are calculated up to n=7
| |
a and normalization integrals rst d3r up ton=5.
n,m
J | | R
4 PACSnumbers: 31.15.-p,71.15.R-m,02.30.Mv
2 Keywords: GaussianTypeOrbitals,GTO,Transformation,BasisFunctions, MatrixElements

]
h I. SCOPE z2. It is normalized as56
p
- π 3/2 3
m GaussianType Orbitals (GTO’s)are widely usedcon- g2(n1,n2,n3)d3r= αn (2nj 1)!!. (2)
2α −
e struction elements of basis sets in quantum chemistry. Z (cid:16) (cid:17) jY=1
h
Theybecamehighlysuccessfulowingtofairlysimplean-
(n+1)(n+2)/2 different g(n ,n ,n ) exist for a given
c 1 2 3
alytical representations of key integrals1,2,3,4,5 that pay
. n. If n 2, they build an overcomplete set of states
cs offintermsofspeedwhenthematrixelementsarecalcu- compared≥to only 2n+1 eigenstates Ynm of the angular
i lated. This work deals with the conversion of Cartesian momentum operator.
s or Hermite GTO’s which have a product representation
y
in Cartesian coordinates into harmonic oscillator func- 2l+1(l m)! 1/2
ph tions which have a product representation in spherical Ylm(θ,ϕ)≡ 4π (l−+|m|)! Pl|m|(cosθ)eimϕ (3)
[ coordinates. The latter are orthogonal with respect to (cid:26) | | (cid:27)
two quantum indices — the total number of integrals to ( l m l)shalldenote SphericalHarmonicsinspher-
4 becalculatedisreducedbytheoverlapandkineticenergy ic−al≤coord≤inates,78 and
v
1 integrals over products of orbitals with the same center. dm
Pm(u) (1 u2)m/2 P (u) (4)
5 Sec. II introduces part of the notation. Sec. III lists l ≡ − dum l
0 those linear combinations of Gaussians defined in Carte-
7 generalized Legendre Polynomials. Their real-valued
siancoordinatesthatobtaintheorthogonalpropertyand
0 counterparts are
are found to be the harmonic oscillator eigenfunctions.
9
9 Toextendcoverageofthefunctionalspace,theoscillator Ylm,c ≡ (Ylm+Yl−m)/√2∝Plm(cosθ)cos(mϕ), (5)
cs/ ftuwnoc-tcieonntseraroevegrelnaepr,akliinzeedticinenSeercg.yIVa.ndSeCctoiuolnomVbcionmtepgurtaelss Ylm,s ≡ −i(Ylm−Yl−m)/√2∝Plm(cosθ)sin(mϕ),(6)
i of those without recourseto the expansions in Cartesian and
s
y coordinates. Sec. VI tabulates some of their products in Y0,c Y0 P0(cosθ), (7)
h supportof2-particleCoulombIntegrals. Absolutenorms l ≡ l ∝ l
p of orbitals play some role in Fitting Function techniques (0 < m l), all normalized to unity, with parity ( )l,
: and are given in Sec. VII for the types of Gaussians dis- ≤ −
v and orthogonal,
i cussed before.
X π 2π
sinθdθ dϕYm,c(θ,ϕ)Ym′,c(θ,ϕ)
r l l′
a Z0 Z0
π 2π
II. TERMINOLOGY = sinθdθ dϕYm,s(θ,ϕ)Ym′,s(θ,ϕ)
l l′
Z0 Z0
=δ δ ; (8)
ll′ mm′
Let
π 2π
sinθdθ dϕYm,c(θ,ϕ)Ym′,s(θ,ϕ)=0. (9)
l l′
g(n1,n2,n3)≡αn/2Hn1(√αx)Hn2(√αy)Hn3(√αz)e−αr2 Z0 Z0
(1) The main result of this work is support to use of GTO’s
be a primitive Hermite GTO (HGTO) with exponent α in systems with heavy or highly polarized atoms by re-
and quantum number n n +n +n centered at the ductionoftheovercompletesetstosetsof2n+1linearly
1 2 3
origin. H areHermiteP≡olynomials,andr2 x2+y2+ independent combinations of GTO’s.
ni ≡
2
III. COMPLETE SETS OF GAUSSIAN TYPE t3,3 =g(3,0,0) 3g(1,2,0)...6;
−
ORBITALS
t =8g(0,0,4)+6g(2,2,0) 24(g(2,0,2)+g(0,2,2))
4,0
−
+3(g(4,0,0)+g(0,4,0))...1680;
A. Hermite Basis
t =4g(1,0,3) 3g(1,2,1) 3g(3,0,1)...42;
4,1
− −
Below,HGTO’sarelinearlycombinedintosetsofreal- t4,2 =6(g(2,0,2) g(0,2,2))
−
valued functions t (r,θ,ϕ,α) ( n m n) that dis- g(4,0,0)+g(0,4,0)...84;
n,m
apnladytangulaYr demp,se(nθd,eϕn)cifeosrmtn,m<0∝−bYynm≤w,acy(θo,≤fϕc)onfosrtrmuct≥ion0 t4,−2 =6g−(1,1,2)−g(1,3,0)−g(3,1,0)...21;
(see An,pmpe∝ndinx−A). Hence they are orthogonal9 t4,3 =g(3,0,1)−3g(1,2,1)...6;
t =g(4,0,0)+g(0,4,0) 6g(2,2,0)...48;
4,4
−
π 2π
t =g(3,1,0) g(1,3,0)...3;
sinθdθ dϕtn,mtn′,m′ δn,n′δm,m′ (10) 4,−4 −
Z0 Z0 ∝ t5,0 =8g(0,0,5)+15(g(4,0,1)+g(0,4,1))
andcompletewithrespecttotheangularvariables. Their 40(g(2,0,3)+g(0,2,3))+30g(2,2,1)...15120;
−
norms are listed in form of t =g(5,0,0)+2g(3,2,0) 12g(3,0,2)+g(1,4,0)
5,1
−
+8g(1,0,4) 12g(1,2,2)...1008;
t2 d3r =N √2π3α 3αn. (11) −
n,m nm − t =2(g(2,0,3) g(0,2,3))
5,2
Z −
g(4,0,1)+g(0,4,1)...36;
The Nnm follow each time after three dots and are eas- −
ily derived from the expansion coefficients given and the t5, 2 =2g(1,1,3) g(3,1,1) g(1,3,1)...9;
− − −
overlap integrals610 t =8g(3,0,2) g(5,0,0)+2g(3,2,0)
5,3
−
+3g(1,4,0) 24g(1,2,2)...864;
g(n1,n2,n3)g(n′1,n′2,n′3)d3r t =g(4,0,1)+g−(0,4,1) 6g(2,2,1)...48;
5,4
Z −
= α(n+n′)/2(cid:16)2πα(cid:17)3/2jY=31(−)⌊n2j⌋+⌊n2′j⌋(nj +n′j −(11)2!!) t5,5t=g=(t551,,−604g,(0=0),−g0(,1360,)1g,(135),(2−g,(0g0)(,+16,,350,g)1(+1),.g4.(,.630,;)0.,.0.)4)80;
( ,allnj +n′jeven 6,0 −
0 ,anyn +n odd 120(g(0,2,4)+g(2,0,4))+90(g(0,4,2)+g(4,0,2))
j ′j −
+180g(2,2,2) 15(g(2,4,0)+g(4,2,0))...665280;
using Eq. (7.374.2) of Ref. 11 or Eqs. (20) and (21) of −
t =8g(1,0,5) 20g(1,2,3)+5g(1,4,1)
Ref. 12. The radial dependence tn,m (αr)nexp( αr2) 6,1 −
∝ − 20g(3,0,3)+10g(3,2,1)+5g(5,0,1)...7920;
is found by inspection of the integralsmentionedin item
−
1 of App. A, which leads to t =g(6,0,0) g(0,6,0)+g(4,2,0)
6,2
−
g(2,4,0)+16(g(0,4,2) g(4,0,2))
πN − −
tn,m =2n+2s(2n+n,1m)!!(αr)ne−αr2Ynm{c,s}, (13) +16(g(2,0,4)−g(0,2,4))...12672;
t =16g(1,1,4) 16(g(1,3,2)+g(3,1,2))
6, 2
− −
where +g(1,5,0)+g(5,1,0)+2g(3,3,0)...3168;
t =8g(3,0,3) 24g(1,2,3)+9g(1,4,1)
Ynm{c,s} ≡(cid:26)YYn−nmm,c,s,, mm≥<00 . (14) 6,3 +6g(3,2,1)−−3g(5,0,1)...3168;
t =10(g(0,4,2)+g(4,0,2))
6,4
+5(g(2,4,0)+g(4,2,0)) 60g(2,2,2)
t =g(0,0,0)...1/4; −
0,0 g(0,6,0) g(6,0,0)...10560;
t =g(0,0,1)...1/4; − −
1,0 t =g(1,5,0) g(5,1,0)
6, 4
t =g(1,0,0)...1/4; − −
1,1 +10(g(3,1,2) g(1,3,2))...660;
t =2g(0,0,2) g(2,0,0) g(0,2,0)...3; −
2,0 − − t6,5 =g(5,0,1)+5g(1,4,1) 10g(3,2,1)...480;
t =g(1,0,1)...1/4; −
2,1 t =g(6,0,0) g(0,6,0)
6,6
t =g(2,0,0) g(0,2,0)...1; −
2,2 − +15(g(2,4,0) g(4,2,0))...5760;
t =g(1,1,0)...1/4; −
2, 2 t =3(g(1,5,0)+g(5,1,0)) 10g(3,3,0)...1440;
− 6, 6
t =2g(0,0,3) 3(g(0,2,1)+g(2,0,1))...15; − −
3,0 − t7,0 =16g(0,0,7) 35(g(0,6,1)+g(6,0,1))
t =4g(1,0,2) g(3,0,0) g(1,2,0)...10; −
3,1 − − 105(g(2,4,1)+g(4,2,1)) 168(g(0,2,5)+g(2,0,5))
t =g(2,0,1) g(0,2,1)...1; − −
3,2 − +210(g(0,4,3)+g(4,0,3))+420g(2,2,3)...8648640;
t =g(1,1,1)...1/4;
3, 2
−
3
t7,1 =240g(3,2,2) 5g(1,6,0)+64g(1,0,6) B. Cartesian Basis
−
240g(1,2,4)+120g(1,4,2) 240g(3,0,4) 5g(7,0,0)
− − −
Thepreviouslistcontainsalsotheexpansionsinterms
15g(3,4,0)+120g(5,0,2) 15g(5,2,0)...4942080;
− − of Cartesian GTO’s (CGTO’s) as outlined in App. B. If
t =15(g(6,0,1) g(0,6,1))
7,2 − each g(n1,n2,n3) in the list is replaced by the primitive
+15(g(4,2,1) g(2,4,1))+48(g(2,0,5) g(0,2,5)) CGTO of the same triple index,
− −
+80(g(0,4,3) g(4,0,3))...823680;
t7, 2 =48g(1,1,5)− 80(g(1,3,3)+g(3,1,3)) f(n1,n2,n3)≡xn1yn2zn3e−αr2, (18)
+15(−g(1,5,1)+g(5,1−,1))+30g(3,3,1)...205920; and each tn,m by t˜n,m, the newly defined t˜n,m are
rnexp( αr2)Ym,c (m 0) and rnexp( αr2)Y m∝,s
t7,3 =3g(7,0,0) 240g(1,2,4)+180g(1,4,2) − n ≥ ∝ − n−
− (m<0), for example
9g(1,6,0)+80g(3,0,4)+120g(3,2,2)
15g(−3,4,0) 60g(5,0,2) 3g(5,2,0)...1647360; t˜4,1 = 4f(1,0,3) 3f(1,2,1) 3f(3,0,1)
− − − − −
t7,4 =601g0((2g,(02,,43,)3+)+15g(g(4(2,0,4,3,1)))−3g((g4(,62,,01,)1))..+.3g7(404,60,;1)) = 43r25πr4e−αr2Y41,c(θ,ϕ). (19)
− −
t =3(g(1,5,1) g(5,1,1)) The normalization integrals are recoveredfrom the Kro-
7, 4
− −
+10(g(3,1,3) g(1,3,3))...2340; neckerproductofthevectorsoftheexpansioncoefficients
− and the overlap integrals6
t =60g(1,4,2) g(7,0,0) 5g(1,6,0) 120g(3,2,2)
7,5
− − −
+5g(3,4,0)+12g(5,0,2)+9g(5,2,0)...149760; f(n ,n ,n )f(n ,n ,n )d3r
1 2 3 ′1 ′2 ′3
t7,6 =g(6,0,1) g(0,6,1) Z
− 3
+15(g(2,4,1) g(4,2,1))...5760; 1 π 3/2
t =3(g(1,5,1)+g(−5,1,1)) 10g(3,3,1)...1440; (4α)(n+n′)/2 2α (nj +n′j −1)!!
7,−6 − = (cid:16) (cid:17) jY=1 (20)
t7,7 =g(7,0,0) 7g(1,6,0)+35g(3,4,0) ( ,allnj +n′jeven
−
0 ,anyn +n odd.
21g(5,2,0)...80640; j ′j
−
t8,0 =128g(0,0,8)+35(g(0,8,0)+g(8,0,0)) to become
+3360(g(0,4,4)+g(4,0,4)) 1792(g(0,2,6)+g(2,0,6))
− t˜2 d3r =N √2π3α 3/(4α)n. (21)
1120(g(0,6,2)+g(6,0,2))+6720g(2,2,4) n,m nm −
− Z
3360(g(2,4,2)+g(4,2,2))+140(g(2,6,0)+g(6,2,0))
− [The similarity between Eq. (12) and Eq. (20) lets the
+210(4,4,0)...8302694400; N alreadygiveninEq.(11)showup again. The addi-
nm
t =g(8,0,0)+g(0,8,0) 28(g(6,2,0)+g(2,6,0)) tional sign in Eq. (12) is positive if applied to g-terms of
8,8
− a single t , and does not mix things up.]
+70g(4,4,0)...1290240; n,m
A synopsis of Eqs. (11), (13) and (21) demonstrates
t =g(7,1,0) g(1,7,0)
8, 8
+7(g(−3,5,0) g(5,3−,0))...20160. t˜n,m =tn,m/(2α)n. (22)
−
An explicit expansion of the vector harmonics multi-
t with odd m are not shown explicitly, but are in-
n, m
cor−porated implicitly by an interchange of the first two plied with the Gaussian in terms of CGTO’s is derived
in App. C:10
arguments of every g on the right hand side of t and
n,m
multiplication by ( 1) m/2 , like for example
⌊ ⌋ 1 2n+1
− rne αr2Ym c,s = (n m)!(n+ m)!
t4, 1 =4g(0,1,3) 3g(2,1,1) 3g(0,3,1)...42. (15) − n { } 2mr 2π −| | | |
− − − σ1+σ2≤⌊(n−|m|)/2⌋ 1 1 σ1+σ2 1
This follows from applying the mirror operation x y
tcoootrhdeineaqtueas,tiaonnds sinhdowucne,swchosic(hmϕis)ϕ ↔(π1/)2m−/2ϕ sinin(pm↔olϕa)r × σ1X,σ2≥0 σ1!σ2!(cid:18)−4(cid:19) (|m|+σ1+σ2)!
and Ym,c ( 1) m/2 Ym,s if m is↔odd−. ⌊ ⌋ 1 |m| m
n ↔ − ⌊ ⌋ n ( )⌊j/2⌋ | |
Application of the Laplace operator yields × (n m 2(σ +σ ))! − j
−| |− 1 2 j=X0(1) (cid:18) (cid:19)
2t =2α(2αr2 2n 3)t . (16) f(m j+2σ ,j+2σ ,n m 2(σ +σ )). (23)
n,m n,m 1 2 1 2
∇ − − × | |− −| |−
The t with exponential α = Mω/(2h¯) are eigenfunc- (In case of Ym,c the sum over j attains only even values
n,m n
tions of the 3-dimensional isotropic harmonic oscillator of j, in case of Ym,s only odd values.) This triple sum
n
is t˜ up to a normalization factor that depends on n
n, m
( ¯h2 2+ 1Mω2r2)t =(n+ 3)h¯ωt . (17) andm±,andtherefore(uptoadifferentfactor)alsotn, m
−2M∇ 2 n,m 2 n,m when f is replaced by g. ±
4
C. Rayleigh-Type Expansion All subsequent considerations assume that s is a non-
negative even integer. (i) This ensures that t is an-
n,m,s
other linear combination of HGTO’s, which is made ex-
The expansion of a plane wave in terms of the t
n,m
plicit in App. D. Therefore a standard, indirect path
reads
of computing integrals is already established: linkage to
eik·r = 2e−k2/(4α) ∞ ik n kthneowinnteagprpanrodasc.heAs1,r2e,3le,4vabnyt daipspsolicciaattiioonniosfgaivllentni,nm,sSeicn.
α
n=0(cid:18) (cid:19) VI. (ii)Therestrictionleadstotruncationofsomeseries
X
n 2π expressions that follow — important to numerical eval-
× m= ns(2n+1)!!NnmYnm{c,s}(k)tn,m(,24) useartiieosn. —(iiiw)hse+reans>odd3s/w2oiusldneseodmedettimoegsuyairealndteineficnointe-
X− −
b vergence of the normalization integral, and (iv) the set
similar to the Rayleigh expansion
of s = 0,2,4,... suffices to let the t span the entire
n,m,s
vector space of the GTO’s.
l
eikr =4π ∞ ilj(kr)Ym (kˆ)Ym(ˆr), (25) The generalization of Eq. (11) reads
· l l ∗ l
l=0m= l
X X− (2s+2n+1)!!
t2 d3r =N √2π3α 3αn, (29)
whererdenotestheangularvariablesofthevectorr. The n,m,s nm(2n+1)!!(4α)s −
Z
expansion coefficients are the quotients of the integrals
eikr t / t t . If r points into the z-direction,
· bn,m n,m n,m and the generalization of Eq. (16)
h | i h | i
we have
1 3
Ynm{c,s}(θk =0,ϕk)=r2n4+π 1δm,0, (26) ∇2tn,m,s(α,r) = hs(2n+s+1)r2 −4α(cid:18)s+n+ 2(cid:19)
+4α2r2 t (α,r). (30)
whence n,m,s
i
2
eikz =√2e k2/(4α) 2t +2ikt 1 k t tn,m,s is an eigenfunction of a spherical potential with
− ( 0,0 α 1,0− 3(cid:18)α(cid:19) 2,0 repulsive core as detailed in App. E.
i k 3 1 k 4 i k 5
t + t + t
3,0 4,0 5,0
−15(cid:18)α(cid:19) 420(cid:18)α(cid:19) 3780(cid:18)α(cid:19) ···)
B. Fourier (Momentum) Representation
=√2e−k2/(4α) ∞ 1 ik ntn,0. (27)
nX=0 (2n−1)!!Nn0 (cid:18)α(cid:19) The Fourier integralof the orbital centered at the ori-
ginisrelatedtoEq.(B2). TheRayleighexpansion(25)is
p
The closely related Cartesian case is written down by inserted,then the radialintegralis solvedandre-written
replacing tn,0 with (2α)nt˜n,0 in Eq. (27) for all n. with Eqs. (11.4.28) and (13.1.27) of Ref. 13:
IV. GENERALIZED OSCILLATOR FUNCTIONS tn,m,s(α,k) eik·rtn,m,s(α,r)d3r
≡
Z
N Γ(n+ 3 + s)
A. Definition = 4π2 n,m (ik)n 2 2
s(2n+1)!! Γ(n+ 3)α(3+s)/2
2
The basis of the t defined in the previous sections s 3 k2
is incomplete with rens,pmect to the radial coordinate. The ×e−k2/(4α) 1F1(−2;n+ 2;4α) Ynm{c,s}(k()3,1)
combination g(2,0,0)+g(0,2,0)+g(0,0,2) of HGTO’s
containsacomponent∝r2e−αr2Y00,c,forexample,which s/2| (−s/2{)zσ k2} σ b
cannot be represented in terms of tn,m. The subsequent σ=0σ!(n+3/2)σ (cid:18)4α(cid:19)
sections therefore introduce a more generaltype of func- X
tions centered at space points A, which is derived from N (s/2)!
Eq. (13) by multiplication with rs: = 4π2s(2n+n,m1)!!(ik)nα(3+s)/2
k2
πN e k2/(4α)L(n+1/2) Ym c,s (k),
t (α,r A) 2n+2 n,m r As(αr A)n × − s/2 4α n { }
n,m,s − ≡ s(2n+1)!!| − | | − | (cid:18) (cid:19)
b
e α(r A)2Ym c,s (r A). (28) where Pochhammer’s Symbol14 has been used.
× − − n { } −
d
5
V. INTEGRALS OF OSCILLATOR FUNCTIONS +( )m l n n′ Y0,c(Cˆ); (35)
− 0 m m′ l
(cid:18) − (cid:19)
A. Overlap Integral m<m <0:
′
The overlapintegralis a convolutionin realspace and y¯lmnnm′′ =−(−)√m2+m′ m+l m′ nm nm′′ Yl|m+m′|,c(Cˆ)
calculated with the shift theorem of Fourier analysis by (cid:18) − − (cid:19)
use of Eq. (31) with tn,m,s(α,k) and tn′,m′,s′(β,−k): +(−√)2m m l m′ nm mn′′ Yl|m−m′|,c(Cˆ); (36)
(cid:18) − − (cid:19)
tn,m,s(α,r A)tn′,m′,s′(β,r B)d3r m=m′ <0:
− −
= Z tn,m,s(α,r)tn′,m′,s′(β,r−C)d3r y¯lmnnm′′ =−√12(cid:18)m+l m′ −nm −nm′′ (cid:19)Yl|m+m′|,c(Cˆ)
Z d3k +( )m l n n′ Y0,c(Cˆ); (37)
= (2π)3e−ik·Ctn,m,s(α,k)tn′,m′,s′(β,−k) − (cid:18)0 −m m′ (cid:19) l
Z m> m′ ,m′ <0:
| |
= 2n+n′(4π)2s(2n+1)!π!(N2nn,′m+N1n)′!,!mα′s+3βs′+3e−γC2 y¯lmnnm′′ = (−)√m2+m′ m+l m′ nm nm′′ Ylm+|m′|,s(Cˆ)
(cid:18) | | − −| |(cid:19)
γ(n+n′+3)/2Γ n+ 3 + s Γ n′+ 3 + s′ (−)m l n n′ Ym−|m′|,s(Cˆ); (38)
× (cid:18) 2 2(cid:19) (cid:18) 2 2(cid:19) − √2 (cid:18)m−|m′| −m |m′|(cid:19) l
s/2 ( s/2) γ σ s′/2 ( s/2) γ σ′ 0<m m′ ,m′ <0:
σ ′ σ′ ≤| |
× σX=0σ!Γ(n−+ 23 +σ)(cid:16)α(cid:17) σX′=0σ′!Γ(−n′+ 32 +σ′)(cid:18)β(cid:19) y¯lmnnm′′ = (−)√m2+m′ m+l m′ nm nm′′ Ylm+|m′|,s(Cˆ)
n+n′ (cid:18) | | − −| |(cid:19)
× (−)(n−n′−l)/2(γC2)l/2Γ(23 + n+n2′+l +σ+σ′) +(−)m′ l n n′ Y−m+|m′|,s(Cˆ). (39)
l=|Xn−n′| √2 (cid:18)m−|m′| −m |m′|(cid:19) l
ymm′(Cˆ)(n+n′−l)/2+σ+σ′ (l−n2−n′ −σ−σ′)σ˜(γC2)σ˜. [Occurrences of Yn0,s have to be interpreted as 0 in Eq.
× lnn′ σ˜!Γ(l+ 3 +σ˜) (39).] Some cases like m<0,m′ >0 or m′ <m<0 are
σX˜=0 2 notcoveredbyEqs.(33)–(39),butmaybedispatchedby
(32)
help of the permutation symmetries
lHaerreintCegr≡al yBm−m′Aisaenxdprγess≡edαinβ/t(eαrm+sβo)f.WTighneera’sng3uj-- ylmnnm′′(Cˆ)=ylmn′′nm(Cˆ) ; y¯lmnnm′′(Cˆ)=y¯lmn′′mn(Cˆ). (40)
lnn′
Symbol:1516 ( 33)17 (Chap. XIV) In case of common center (C = 0), the orthogonality
§
relation18 ( (4b))
l §
ymm′(Cˆ) Yκ (Cˆ) sinθ dθdϕ
lnn′ ≡ l ∗ k k t (α,r A)t (β,r A)d3r
κ= l Z n,m,s n′,m′,s′
X− − −
×Ylκ(kˆ)Ynm{c,s}(kˆ)Ynm′′{c,s}(kˆ) Z=δ δ 22n+3πNn,mγnΓ(n+ 23 + 2s + s2′)(41)
= (2l+1)(2n+1)(2n′+1) l n n′ y¯mm′(Cˆ). n,n′ m,m′ (2n+1)!! (α+β)(s+s′+3)/2
4π 0 0 0 lnn′
r (cid:18) (cid:19) results,whichturns into Eq.(29) ifα=β ands=s. [If
′
Eq. (4.6.3) of Ref. 15 and adaptation to its sign conven- thederivationstartsfromEq.(32),theterm becomes
σ˜
tion of Spherical Harmonics determine y¯mm′(Cˆ): 1/Γ(l+3/2). Thefactor(γc2)l/2thenmeansthatnonzero
lnn′ P
overlapcan occur only if the term l=0 is present in the
m =0:
′ sum over l. In turn, l = 0 becomes the only term that
y¯lmnnm′′ =(−)m(cid:18) ml −nm n0′ (cid:19)Ylm{c,s}(Cˆ); (33)cinotnrtordibuucteess,aafnadctionrspδnecnt′.ionThoefnthye0mnlomnw′e=rlδimmimt′l/=(4π|n),−ann′d|
m>m′ >0: eventually
y¯lmnnm′′ = (−)√m2+m′ m+l m′ nm nm′′ Ylm+m′,c(Cˆ) s/2 ( s/2)σ γ σ s′/2 ( s′/2)σ′ γ σ′
(cid:18) − − (cid:19) − −
+(−√)2m m l m′ nm mn′′ Ylm−m′,c(Cˆ); (34) σX=0σ!Γ(n3+ 23 +σ)(cid:16)α(cid:17) σX′=0σ′!Γ(n+ 23 +σ′)(cid:18)β(cid:19)
(cid:18) − − (cid:19) Γ(n+ +σ+σ )
′
m=m′ >0: × 2
y¯lmnnm′′ = √12(cid:18)m+l m′ −nm −nm′ ′ (cid:19)Ylm+m′,c(Cˆ) = Γ(n1+ 23)F2(n+ 23,−2s,−s2′,n+ 23,n+ 23;αγ,βγ)
6
3 s 3 s
γn+5/2Γ n+ + Γ n+ + ′
= γ s/2 γ s′/2 Γ(n+ 23 + 2s + s2′) (42) × (cid:18) 2 2(cid:19) (cid:18) 2 2(cid:19)
(cid:18)β(cid:19) (cid:16)α(cid:17) Γ(n+ 23 + 2s)Γ(n+ 32 + s2′) s/2 (−s/2)σ γ σ s′/2 (−s′/2)σ′ γ σ′
× σ!Γ(n+ 3 +σ) α σ !Γ(n+ 3 +σ ) β
is calculated with Eqs. (9.180.2), (9.182.3) and (9.122.1) σX=0 2 (cid:16) (cid:17) σX′=0 ′ 2 ′ (cid:18) (cid:19)
of Ref. 11.] 5
Γ n+ +σ+σ . (45)
′
× 2
(cid:18) (cid:19)
B. Kinetic Energy Integral
C. 1-Particle Coulomb Integral
Two “direct” methods exist to calculate the kinetic
energy matrix. (i) Application of the kinetic en-
The Fourier technique also suits the attack on the
ergy operator on t (β,r B) yields three terms
n′,m′,s′ Coulomb integral:
tn′,m′,s′ 2(β,r B), tn−′,m′,s′(β,r B) and
∝tbny′,Emq′,.s′(+126()β−i,frs−=B0−).bKyinEeqt∝.ic(3en0)erigfys′in6=te0g,raa−lnsdthtuwsorteedrumc∝es tn,m,s(α,r−A) r 1r tn′,m′,s′(β,r−B)d3rd3r′
′ Z | − ′|
to two- or threefold application of Eq.(32);1918 common d3k e ikC
factors like ylmnnm′′ could be reused. (ii) The second di- = 4π (2π)3 −k2· tn,m,s(α,k)tn′,m′,s′(β,−k)
rect method is based on the conversion of the Laplace Z
operator to a squared wave number in Fourier space: = 2n+n′16π3 πNn,mNn′,m′ e γC2
s(2n+1)!!(2n′+1)!!αs+3βs′+3 −
t (α,r A) 2t (β,r B)d3r
Z n,m,s − ∇r n′,m′,s′ − γ(n+n′+1)/2Γ n+ 3 + s Γ n′+ 3 + s′
d3k × 2 2 2 2
= −Z (2π)3e−ik·Ck2tn,m,s(α,k)tn′,m′,s′(β,−k)(.43) s/2 ( s/2(cid:18))σ γ σ(cid:19)s′/2(cid:18) ( s′/2)σ(cid:19)′ γ σ′
− −
The common result of both approaches may be summa- × σ!Γ(n+ 3 +σ) α σ !Γ(n + 3 +σ ) β
rized as σX=0 2 (cid:16) (cid:17) σX′=0 ′ ′ 2 ′ (cid:18) (cid:19)
n+n′ Γ(3 + n+n′+l +σ+σ 1)
2 ( )(n n′ l)/2(γC2)l/2 2 2 ′−
Z tn,m,s(α,r−A)(cid:18)−∇2r(cid:19)tn′,m′,s′(β,r−B)d3r ×l=|Xn−n′| − − − Γ(l+ 32)
l n n 3
= 2n+n′+1(4π)2s(2n+1)!π!(N2nn,′m+N1n)′!,!mα′s+3βs′+3e−γC2 × ylmnnm′′(Cˆ)1F1( − 2− ′ −σ−σ′+1;l+ 2;γC2). (46)
The limit of common center is
3 s 3 s
× γ(n+n′+5)/2Γ n+ 2 + 2 Γ n′+ 2 + 2′ t (α,r A) 1 t (β,r A)d3rd3r
(cid:18) (cid:19) (cid:18) (cid:19) n,m,s − r r n′,m′,s′ − ′
s/2 (−s/2)σ γ σ s′/2 (−s′/2)σ′ γ σ′ = δZ δ 22|n+−3π2′N| n,m
× σX=0σ!Γ(n+ 23 +σ)(cid:16)α(cid:17) σX′=0σ′!Γ(n′+ 32 +σ′)(cid:18)β(cid:19) n,n′ m,m′(2n+1)!!α(s+3)/2β(s′+3)/2
n+n′ ( )(n−n′−l)/2(γC2)l/2 × γn+1/2Γ n+ 23 + 2s Γ n+ 23 + s2′
× − (cid:18) (cid:19) (cid:18) (cid:19)
l=|Xn−n′| s/2 ( s/2) γ σ s′/2 ( s/2) γ σ′
× Γ(cid:18)23 + n+n2′+l +σ+σ′+1(cid:19)ylmnnm′′(Cˆ) × σX=0σ!Γ(n−+ 23 σ+σ)(cid:16)α(cid:17) σX′=0σ′!Γ(−n+′ 23 σ+′ σ′)(cid:18)β(cid:19)
(n+n′−l)/2+σ+σ′+1(l−n2−n′ −σ−σ′−1)σ˜(γC2)σ˜, (44) × Γ n+ 21 +σ+σ′ . (47)
× σ˜!Γ(l+ 3 +σ˜) (cid:18) (cid:19)
σ˜=0 2
X In the general case of l < n + n or σ + σ > 0, the
′ ′
which is Eq. (32) multiplied by 2γ plus the replacement representation
σ+σ σ+σ +1 at three places in each of the last
two lin′e→s of both′equations. The limit of commoncenter 1F1(l−n−n′ σ σ′+1;l+ 3;γC2)
2 − − 2
(C =0) is
tn,m,s(α,r−A) −∇22r tn′,m′,s′(β,r−A)d3r =(n+n′−l)σ˜/=2+0σ+σ′−1(l−n2−nσ˜′!−(l+σ−23)σσ˜′+1)σ˜(γC2)σ˜(48)
Z (cid:18) (cid:19) X
22n+4πNn,m lets Eq. (46) look like Eq. (32) multiplied by π/γ plus
= δ δ
n,n′ m,m′(2n+1)!!α(s+3)/2β(s′+3)/2 the replacement σ + σ′ σ + σ′ 1 at three places.
→ −
7
The special case of l = n + n = σ + σ = 0 may be where C = B A = (C ,C ,C ). Combined at P,
′ ′ x y z
−
paraphrasedwith Eq. (7.1.21) of Ref. 13, the product is a sum of g of quantum numbers up to
n+n +s+s, and finally re-converted into a sum over
F (1;3,γC2)= π eγC2erf γC2. (49) t.,.,.(α′+β,r P′ )asdemonstratedinApp.A. Tosimplify
1 1 2 γC2 2 theintermed−iatesteps,oneactuallyusesCGTO’sandthe
r
p associated
The remaining cases of l = n+n > 0,σ+σ = 0 could
′ ′
be recursively attached to the representation (49) with t˜ rst˜ =t /(2α)n (54)
n,m,s n,m n,m,s
Eq. (13.4.6) of Ref. 13, ≡
to work this out. Switching from the t˜representation to
3 l+1/2 1
F (1;l+ ,γC2)= F (1;l+ ,γC2) 1 . the f representation is done with the coefficients of the
1 1 2 γC2 1 1 2 −
(cid:26) (cid:27) table in Sec. IIIA. To switch back one builds a table
(50)
with the projection technique of App. A that starts as
Some basic observations on the numerical evaluation
follows:
of Eqs. (32), (44) and (46) are:
ymm′ =0ifl+n+n isodd. Thereforeeachsecond f(0,0,0)=t˜0,0,0;
• telnrmn′ in the sum ove′r l may be skipped. f(0,0,1)=t˜1,0,0;
f(0,1,0)=t˜ ;
All appearances of the Γ-function have the form 1,−1,0
• Γ(j+3/2)=(2j+1)!!√π/2j+1 with smallintegers f(1,0,0)=t˜1,1,0;
j,andmaybe placedina lookuptable. (The asso- f(0,0,2)= 1(t˜ +t˜ );
3 2,0,0 0,0,2
ciated factors √π may be dropped, since the same f(0,2,0)= 1t˜ 1t˜ + 1t˜ ;
number of Γ’s appears in numerators and denomi- −6 2,0,0− 2 2,2,0 3 0,0,2
nators.) f(2,0,0)=−61t˜2,0,0+ 12t˜2,2,0+ 31t˜0,0,2;
f(0,1,1)=t˜ ;
Ingeneral,the numberoftermsinthe sumsiscon- 2, 1,0
−
• siderably smaller than the number of GTO pairs f(1,0,1)=t˜2,1,0;
in the “dissociated” products. The focus shifts to f(1,1,0)=t˜ ;
2, 2,0
e(3ffi9c)i.ent implementation of the y¯lmnnm′′ in Eqs. (33)– f(0,0,3)= 51t˜3,0,0+− 53t˜1,0,2;
f(0,3,0)= 1t˜ 3 t˜ + 3t˜ ;
−4 3,−3,0− 20 3,−1,0 5 1,−1,2
f(0,1,2)= 1(t˜ +t˜ );
VI. TABLE OF PRODUCTS 5 3,−1,0 1,−1,2
f(1,0,2)= 1(t˜ +t˜ );
5 3,1,0 1,1,2
The 2-Particle Coulomb Integral f(0,2,1)=−110t˜3,0,0− 12t˜3,2,0+ 15t˜1,0,2;
t (α,r A)t (α¯,r A¯) 1 f(2,0,1)=−110t˜3,0,0+ 12t˜3,2,0+ 15t˜1,0,2;
n,m,s − n¯,m¯,s¯ ′− r r f(1,2,0)= 1t˜ 1 t˜ + 1t˜ ;
Z | − ′| −4 3,3,0− 20 3,1,0 5 1,1,2
t (β˜,r B˜)t (β,r B)d3rd3r (51) f(2,1,0)= 1t˜ 1 t˜ + 1t˜ ;
× n˜,m˜,s˜ ′− n′,m′,s′ − ′ 4 3,−3,0− 20 3,−1,0 5 1,−1,2
may be computed by (i) expansion of the product f(1,1,1)=t˜3, 2,0;
ttn,m(,αs(+α,βr,r−AP))t,na′,lml′c,se′n(tβe,rred−aBtP) into(αaAsu+mβBof)/t(eαrm+sβ∝), f(0,0,4)= 315t˜4,0,0+ 15t˜−0,0,4+ 72t˜2,0,2;
(·i,i·),·expansio−noftheproductt (≡α¯,r A¯)t (β˜,r f(2,2,0)= 2810t˜4,0,0+ 115t˜0,0,4− 211t˜2,0,2− 81t˜4,4,0.
n¯,m¯,s¯ ′ n˜,m˜,s˜ ′
− −
B˜) likewise into a sum of terms t (α¯+β˜,r P˜), all Examples of products obtained with a Maple20 imple-
,,
centered at P˜ (α¯A¯ +β˜B˜)/(α¯∝+β˜·)·,·and (iii)−comput- mentation follow. The signals the notational conven-
ingtheremaini≡ngsumover1-ParticleCoulombIntegrals tions that the first t˜ on≃the left hand side has the ar-
with centers at P and P˜ as proposed in Sec. VC. guments (α,r A), that the second t˜on the left hand
To implement the first two steps, an expansion of a side has the ar−guments (β,r B), that all t˜on the right
product of two oscillator functions must be at hand. Its handsidehavethearguments−(α+β,r P),andthatthe
evaluation is straight forward as proposed here, though right hand side is to be multiplied by−exp( γC2). The
−
tedious: t (α,r A) and t (β,r B) are both full writing of the first line would be
n,m,s n′,m′,s′
− −
expanded as described in App. D to obtain two sums
t˜ (α,r A)t˜ (β,r B)
of HGTO’s. They are merged with the product rule of 0,0,0 − 0,0,0 −
HGTO’s: Those g are shifted to the common center P =exp( γC2)t˜ (α+β,r P). (55)
0,0,0
− −
β
r A = r P+ C (52) Reducedexponentialsαr α/(α+β)andβr β/(α+β)
− − α+β are also inserted to simpl≡ify the notation. ≡
α
r−B = r−P− α+βC (53) t˜0,0,0t˜0,0,0 t˜0,0,0;
≃
8
t˜ t˜ β2C2t˜ +t˜ ting functions.21,22,23 The measure g(n ,n ,n )d3r =
0,0,2 0,0,0 ≃ r 0,0,0 0,0,2 1 2 3
+2βr(Cxt˜1,1,0+Cyt˜1, 1,0+Czt˜1,0,0); (π/α)3/2δn,0 isgenerallyzeroanddoResnotprovideause-
− ful substitute to Eqs. (2), (11) and (29) for that reason.
t˜ t˜ α2β2C4t˜ +t˜
0,0,2 0,0,2 ≃ r r 0,0,0 0,0,4 The absolute norm as calculated below is the next sim-
4
+(α2+β2 α β )C2t˜ ple alternative. It quantifies how many particles have to
r r − 3 r r 0,0,2 be moved from some region of space to others to real-
+2αrβr(αr βr)C2(Cxt˜1,1,0+Cyt˜1, 1,0+Czt˜1,0,0) ize the specific density relocation,and becomes useful to
− −
2(α β )(C t˜ +C t˜ +C t˜ ) qualifytherelativeimportanceoftermswithproductsof
r r x 1,1,2 y 1, 1,2 z 1,0,2
− − − expansion coefficients and the fitting functions.
2
+3αrβr(C2−3Cz2)t˜2,0,0 Theintegratedabsolutevalueofg(n1,n2,n3)isaprod-
uct of three integrals,
8α β (C C t˜ +C C t˜ +C C t˜ )
r r x y 2, 2,0 x z 2,1,0 y z 2, 1,0
− − −
−2αrβr(Cx2−Cy2)t˜2,2,0; g(n1,n2,n3)d3r =G(n1)G(n2)G(n3), (56)
| |
t˜1,0,0t˜0,0,0 βrCzt˜0,0,0+t˜1,0,0; Z
≃
t˜ t˜ β C t˜ +t˜ ; with
1,1,0 0,0,0 r x 0,0,0 1,1,0
≃
t˜ t˜ β C t˜ +t˜ ;
1, 1,0 0,0,0 r y 0,0,0 1, 1,0
t˜ − t˜ ≃α2β C C2t˜ −+(β 2α )C t˜ G(nj) ≡ ∞ αnj/2|Hnj(√αu)|e−αu2du
1,−1,0 0,0,2 ≃ r r y 0,0,0 r − 3 r y 0,0,2 Z−∞
+α2αr(αβrC(C2−C2t˜βrCy2+)t˜1C,−1C,0+t˜ t˜1,−)1,2 = 2α(nj−1)/2Z0∞|Hnj(u)|e−u2du. (57)
r r x y 1,1,0 y z 1,0,0
−
2α (C t˜ +C t˜ ) G(nj) is a sum of 1+ nj/2 integrals delimited by the
r x 2, 2,0 z 2, 1,0 ⌊ ⌋
− 1 − − positive roots of Hnj. Each integral is solved via Eq.
+α C ( t˜ +t˜ ); (7.373.1) of Ref. 11 or Eq. (22.13.15)of Ref. 13:
r y 2,0,0 2,2,0
3
2
t˜1,0,0t˜0,0,2 ≃α2rβrCzC2t˜0,0,0+(βr− 3αr)Czt˜0,0,2 G(0) = π/α≈1.772453850905516α−1/2;
G(1) = 2;
+α (α C2 2β C2)t˜ +t˜ p
2αrrβrr(Cx−Czt˜1r,1,0z+1C,0y,0Czt˜1,1,01,,20) G(2) = 4 2α/e≈3.431055539842827α1/2;
−2α (C t˜ +C t˜ ) −2α C t˜ ; G(3) = 4αp 1+4e−3/2 ≈7.570082562374877α;
r x 2,1,0 y 2, 1,0 r z 2,0,0
− − − 3 (cid:16) (cid:17)
1 3 √6
t˜1,−1,0t˜1,−1,0 ≃−αrβrCy2t˜0,0,0+ 3t˜0,0,2 G(4) = 4(α/e)3/2 e−√6/2H3s2 + 3 
(cid:20)
1 1
(αr βr)Cyt˜1, 1,0 t˜2,0,0 t˜2,2,0;  
− − − − 6 − 2 3 √6
t˜1,0α,0t˜C1,−t˜1,0 ≃+−βαrCβrt˜CyCzt˜;0,0,0+t˜2,−1,0 −e√6/2H3s2 − 3 (cid:21)
r y 1,0,0 r z 1, 1,0
− − 1 19.855739152211958α3/2;
t˜ t˜ α β C2t˜ + (t˜ +t˜ ) ≈
1,0,0 1,0,0 ≃− r r z 0,0,0 3 2,0,0 0,0,2 G(5) 59.2575529009459587α2;
≈
(αr βr)Czt˜1,0,0; G(6) 195.90006551027769α5/2;
t˜−1,1,0t˜1−, 1,0 αrβrCxCyt˜0,0,0+t˜2, 2,0 G(7) ≈ 704.821503307929489499α3.
α C−t˜ ≃+−β C t˜ ; − ≈
r y 1,1,0 r x 1, 1,0
− − Theintegral f(n ,n ,n )d3riscalculatedwithease
1 1 2 3
t˜1,1,0t˜1,1,0 ≃−αrβrCx2t˜0,0,0+ 3t˜0,0,2 via Eqs. (3.461R.2)| and (3.461.3|) of Ref. 11.
The linear combinations defined in Sec. IIIA may be
1 1
(α β )C t˜ t˜ + t˜ . evaluated in spherical coordinates and decompose into
r r x 1,1,0 2,0,0 2,2,0
− − − 6 2
products of integrals over r [handled by Eq. (3.461) of
Ref. 11], ϕ (yielding 2π or 4 for m = 0 or m = 0, re-
6
spectively) and θ [handled by determining the roots of
VII. ABSOLUTE NORMS Pm(cosθ) and decomposition into sub-intervals]. “Mo-
n
nomic” cases like t , t or t , which relate t to
1,1 2,1 3, 2 n,m
If GTO’s represent orbitals or wave functions, their a single HGTO, are already repr−esented by Eq. (56) and
squares represent particle densities, and normalization not listed again.
follows from integrals like (2) or (11). The first power
rather than the second one specifies local densities, π
t d3r = 8π 25.7190053432553290α 1/2;
if these functions are the constituents of density fit- | 2,0| 3α ≈ −
Z r
9
t d3r = 8 π/α 14.1796308072441282α 1/2; read
2,2 −
| | ≈
Z
Z |t3,0|d3r = 1p04π/5≈65.345127194667699360; Z |tn,m,s(α,r−A)|d3r =α−s/2(cid:18)n+2 3(cid:19)s/2Z |tn,m|d3r.
224 (58)
t d3r = 16arcsin(5 1/2)+4π
3,1 −
| | 5 −
Z
49.94800887034627509;
≈ VIII. SUMMARY
t d3r = 16;
3,2
| |
Z Asetofbasisfunctionshasbeendefinedbyrecombina-
t3,3 d3r = 12π 37.69911184307751886; tion ofHGTO’s or CGTO’s such that the members with
| | ≈
Z commoncenterareorthogonalandcompletewithrespect
192
t d3r = π√35πα (√30+3) 15 2√30 to the angular variables. They turn out to be eigenfunc-
4,0
| | 245 − tions ofthe isotropicharmonicoscillator. Aspecific gen-
Z (cid:16) q
eralization of those allows (i) to keep the computational
+(√30 3) 15+2√30
− advantageof sparse overlapand kinetic energy integrals,
765.99700145937q577804α1/2;(cid:17) (ii) backup by the tabulated GTO’s in case of need for
≈ allintegralsthataremultilinearintheorbitals(2-particle
t d3r = 8√πα 1+128/73/2 Coulomb Integrals), still use of “direct” alternative for-
4,1
| |
Z (cid:16) (cid:17) mulasforoverlapandkineticenergyintegrals,and(iii)to
112.180027342566280α1/2; maintainthevectorspaceoffunctionscoveredbyGTO’s.
≈
t d3r = 32√πα 1+34/73/2
4,2
| |
Z 160.843(cid:16)94454775629(cid:17)92α1/2; Acknowledgments
≈
t d3r = 16√πα 1+34/73/2
4, 2 The work was supported by the Quantum Theory
| − |
Z (cid:16) (cid:17) ProjectattheUniversityofFloridaandgrantDAA-H04-
80.42197227387814961α1/2;
≈ 95-1-0326from the U.S.A.R.O.
t d3r = 24√πα 42.538892421732384655α1/2;
4,3
| | ≈
Z
t d3r = 64√πα 113.4370464579530257α1/2; APPENDIX A: METHOD OF EXPANSION
4,4
| | ≈
Z
t d3r = 16√πα 28.3592616144882564α1/2; The representations in Sec. IIIA were obtained by re-
4, 4
| − | ≈ engineering the straight-forward solution to the inverse
Z
64 problem as follows:
t d3r = π 1701+640√70 α
5,0
| | 567
Z (cid:16) (cid:17) 1. expanding all (n+1)(n+2)/2 GTO’s of a fixed n
2501.97074380428359α;
≈ into the complete set of Ym (0 l n; l m
l ≤ ≤ − ≤ ≤
t5,1 d3r 594.462027151417576α; l). For each triple (n1,n2,n3), this expansion im-
| | ≈ pliesthecalculationof(n+1)2 integralsofthetype
Z
t d3r = 1024α/9=113.¯7α; g(n1,n2,n3)Ylm∗sinθdθdϕ. Most of these vanish
5,2
| | duetotheselectionrules(i)n +n +n +levenand
Z R 1 2 3
(ii) l+m+n even. One mayalsouse thatthe ex-
|t5,−2|d3r = 512α/9=56.¯8α; pansion coeffi3cient in front of Yl−m is the complex
Z conjugate of the one in front of Ym. The pedes-
8α l
t d3r = 3064√2 2916arcsin(1/3)+729π trian’s way to calculate the integral over θ and ϕ
5,3
| | 81 −
Z (cid:16) (cid:17) is (a) transformationof g(n1,n2,n3) into spherical
556.287108307150907α;
≈ coordinates via Eq. (1) and x = rcosϕsinθ etc.
t d3r = 128α; This results in a product of two integrals, one over
5,4
| | θ and one overϕ. (b) The integraloverϕ becomes
Z
elementary if all occurrences of sinϕ and cosϕ are
t d3r = 32α;
| 5,−4| substituted with Euler’s formula. (c) The substi-
Z tution cosθ = u reduces the integral over θ to a
|t5,5|d3r = 120πα≈376.991118430775189α. Hypergeometric Function 3F2 (p. 183 of Ref. 24).
Z
2. gathering and recombining all Ym within each ex-
l
Theabsolutenormsofthegeneralizedoscillatorfunctions pansion in terms of Ym,c and Ym,s. The list at
l l
10
n 3 (partial list at n=4) then reads: and Ym,s with ( ) m/2 Ym,s if m is even.
≤ l − − ⌊ ⌋ l
g(0,0,0) 2Y0,c; 3. selecting the subsetofequationsthat containYnm,c
≃ 0 for fixed m and n, yielding, say, 1 q(m,n)
g(0,0,1)≃ √43RY10,c; (n+1)(n+2)/2equations. [Onemayg≤enerallyfin≤d
g(1,0,0) 4 RY1,c; more stringentupper bounds for q by inspectionof
≃ √3 1 the selection rule (ii) given above.] The remaining
g(0,0,2) (8R2 4)Y0,c+ 16 R2Y0,c; taskistofindq(m,n)numberssuchthatthelinear
≃ 3 − 0 3√5 2
combinationofthese q equationsby these numbers
g(1,0,1)≃ √815R2Y21,c; does not contain any terms Ylm=n,c or Ylm,s. This
g(1,1,0) 8 R2Y2,s; means computing a q-dimensio6nal basis vector of
≃ √15 2 a kernel of a matrix that contains all the expan-
g(2,0,0) (8R2 4)Y0,c 8 R2Y0,c+ 8 R2Y2,c; sion coefficients prior to those Y terms that are to
≃ 3 − 0 − 3√5 2 √15 2
be eliminated. Generally this matrix is non-square
g(0,0,3) 8√3R(2R2 5)Y0,c+ 32 R3Y0,c;
≃ 5 − 1 5√7 3 and r-dependent.
g(1,0,2) 8 R(2R2 5)Y1,c+ 32 2 R3Y1,c; 4. normalizingthisq-dimensionalvectorwithsomear-
≃ 5√3 − 1 5 21 3
g(1,1,1) 16 R3Y2,s;q bitrarinesssuchthatitscomponentsaresmallinte-
≃ √105 3 gersandthattn,m/[(αr)nexp(−αr2)Ynm,c]arepos-
g(2,0,1) 8 R(2R2 5)Y0,c 16 R3Y0,c itive numbers. Those q components are the expan-
≃ 5√3 − 1 − 5√7 3 sion coefficients in front of the g(n ,n ,n ) of the
+√11605R3Y32,c; table form≥0. Countingtermssh1ows2q(63,2)=8,
q(6,0) = 10 and q(3,2) = 2, for example. Experi-
g(2,1,0) 8 R(2R2 5)Y1,s 8 2 R3Y1,s
≃ 5√3 − 1 − 5 21 3 ence shows that — up to this normalization factor
q —theexpansionsareuniqueforn 8atleast,i.e.,
+8 2 R3Y3,s; ≤
35 3 the aforementioned kernel is one-dimensional.
q
g(3,0,0) 8√3R(2R2 5)Y1,c 8 6R3Y1,c 5. performing steps 3 to 4 for the Ym,s in an equiv-
≃ 5 − 1 − 5 7 3 alent manner by elimination of tenrms Ym,s and
+8 325R3Y33,c; q ∝Ylm,c to obtain tn,−m. ∝ l6=n
g(0,0,4) (3q2R4 32R2+24)Y0,c An additional shortcut exists once t and t are
+ 64 R2(≃2R25 7)Y−0,c+ 256R4Y00,c; known for a specific “anchorage” m.nA,mpplicationn,−omf the
7√5 − 2 105 4 ladder operators
g(1,0,3) 16 3(2R2 7)Y1,c+ 64 2R4Y1,c;
≃ 7 5 − 2 21 5 4 L Lx iLy =i(y∂z z∂y) (x∂z z∂x) (A1)
g(1,1,2) 16qR2(2R2 7)Y2,s+ q64 R4Y2,s; ± ≡ ± − ± −
≃ 7√15 − 2 21√5 4 of angularmomentum quantum mechanics on Ynm yields
g(2,0,2)≃(1352R4− 332R2+8)Y00,c Ynm±1. Decomposition of the two equations
+ 16 R2(2R2 7)Y0,c+ 16 R2(2R2 7)Y2,c L (t +it ) t +it (A2)
21√5 − 2 7√15 − 2 ± n,m n,−m ∝ n,m±1 n,−m∓1
128R4Y0,c+ 64 R4Y2,c; into 4real-valuedequationsyields a similarrecursionfor
−105 4 21√5 4
t . Effectively one applies
g(2,1,1) 16 R2(2R2 7)Y1,s 16 2R4Y1,s n,m
≃ 7√15 − 2 − 21 5 4
(y∂ z∂ )g(n ,n ,n )
+136 325R4Y43,s; q =z −n3g(yn1,n21+12,n33 1) n2g(n1,n2 1,n3+1);
− − −
g(2,2,0)≃(1352R4−q332R2+8)Y00,c+ 13025R4Y40,c (x∂z −z∂x)g(n1,n2,n3)
−213√25R2(2R2−7)Y20,c− 3√3235R4Y44,c; =n3g(n1+1,n2,n3−1)−n1g(n1−1,n2,n3+1)
term by term to a pair of equally normalized t and
g(3,0,1) 16 3R2(2R2 7)Y1,c n,m
≃ 7 5 − 2 tn, m.
q T−hequickestalternativetoobtaintheexpansionisde-
16 2R4Y1,c+ 16 2 R4Y3,c.
− 7 5 4 3 35 4 veloped in App. C further down.
q q
The symbol means a factor e αr2αn/2√π has
−
≃
been omitted each time at the right hand side, APPENDIX B: CORRESPONDENCE BETWEEN
and R stands short for √αr. The coefficients of CARTESIAN AND HERMITE GTO’S
g(n ,n ,n ) arederivedfromthose ofg(n ,n ,n )
2 1 3 1 2 3
by replacing Ylm,c with (−)⌊m/2⌋Ylm,c if m is even, A Fourier Transform switches from Cartesian to Her-
Ym,c with( ) m/2 Ym,s andviceversaifmisodd, miteGTO’sandviceversa.25,26AllHGTO’sg(n ,n ,n )
l − ⌊ ⌋ l 1 2 3