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Description of the gauge invariance of molecular magnetic properties of the SF6 molecule
THEO CHEM Journal of Molecular Structure (Theochem) 364 (1996) 69-77
Description of the gauge invariance of molecular magnetic properties of the SF6 molecule M.C. Caputo, M.B. Ferraro*” Departamento
de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria. Buenos Aires, Argentina
Pab. I, (1428)
Received 17 August 1995; accepted 16 November 1995
Abstract Ab initio calculations of the magnetic susceptibility and the nuclear magnetic shielding tensor for the SF, molecule are reported here. Their dependence on the choice of gauge for the vector potential, which describes the external magnetic field, is discussed. The required invariance of molecular magnetic properties depends on the fulfillment of very general gauge-invariance sum rules and on the quality of the basis set employed. Keywords;
Gauge
invariance; Magnetic shielding tensor; Magnetic susceptibility
1. Introduction
The magnetic susceptibility and the nuclear magnetic shielding tensor are molecular properties for which the gauge invariance of the calculated quantities is a physical requirement. A translational transformation of the origin of coordinates may also be interpreted as a gauge transformation of the vector potential. In the past, we have analyzed [l] gauge constraints for different choices of those gauge transformations. For instance, we analyzed exhaustively the Landau transformation [2-41 and the polarization propagator based theory proposed by Geertsen [5] for a series of small molecules: CH4, NH3, HF and HZ0 [6].
* Corresponding author. ’ Member of Carrera de1 Investigador 0166-1280/96/$15.00 SSDZ
de1 CONICET.
In this case we have studied the molecular magnetic properties of the SF6 molecule, which is larger than those analyzed previously by us; the number of electrons to deal with is 70. We evaluated these properties for the Coulomb and the Landau gauges, employing extended Gaussian basis sets to represent the wavefunction, and the Random Phase Approximation (RPA) [7] to describe the perturbation scheme. For the magnetic susceptibility, we also made calculations for two origins of the coordinates and for the different formalisms described in [6] for the diamagnetic susceptibility. The degree to which very general sum rules are fulfilled is a measure of the quality of the basis set employed for determining theoretical molecular properties. General constraints for invariance of magnetic properties in an arbitrary gauge transformation have been analyzed [l] and those corresponding to the transformation from
0 1996 Elsevier Science B.V. All rights reserved
0166-1280(95)04455-8
M.C. Capulo, M.B. Ferraro/Journal
70
of Molecular Structure (Theochem)
Coulomb to Landau gauges were examined in particular. In Section 2 the expressions for the magnetic susceptibility and the nuclear magnetic shielding tensor in the Coulomb and the Landau gauges are given. The sum rules for invariance under a Landau transformation are also included. Section 3 discusses the numerical results for the SF, molecule.
364 (1996) 69-77
been introduced, i.e.,
e2 Ay* HBB =--m,c2 ’
A: = A’(Q)
(7)
If a permanent intrinsic magnetic dipole moment ,u,, placed on nucleus I, with vector potential A’“‘(r) = -pLI x VP Ir -‘&I
2. Landau transformation and its conditions for invariance of magnetic susceptibility and nuclear magnetic shielding tensor
is taken into account, the interaction Hamiltonian contains the additional terms
Let us consider a molecule with ni electrons, with mass m,, charge -e, coordinates ri, canonical moments pi, angular momenta ri = ri x pi (i = 1, 2 . . n), and N nuclei, with corresponding quantitiks ZIe, M,, RI, etc. The “particle” Hamiltonian of the electrons is
e2 e -~(A~‘.p)i+~~(A’.A”)i met i=, m,c2 i=*
(9)
The second order energies wBB = -&+B,B~
= w,“” + wp
‘B p = wt4,B+ wp W MB -_ &a~,/3
(10) (11)
are defined for the Coulomb gauge by
(12) 1 N N Z1ZJe2 +jcc I Jzl I& - RJI
(1)
with eigenstates ]j) and energy eigenvalues E(O); the reference state is denoted by ]u) (the n$ation of related papers [1,8] is retained here, e.g. L, = Cyzl li,; wja = i(Ey’ - Ei’)), etc.). In the presence of a uniform magnetic field B, the divergenceless Coulomb vector potential is a transverse field, defined by the equation A’(r)
= i/l x r
V-A’=0
P
e2 h? c w. 2m2c2A e ~#a Ja
(14)
(2)
(3)
The total Hamiltonian is represented by a truncated Taylor series, H=H”+HB+$HEB
wBB=-
(4)
where the interaction Hamiltonian, containing terms of first and second order in the field, has
(15) In the reference state ]u) the diamagnetic contribution to the magnetic susceptibility tensor is
(16)
M.C. Caputo, M.B. Ferraro/Journal
and the paramagnetic contribution
of Molecular Structure (Theochem)
Therefore, under such gauge transformation
is
ApPIB _ --AT”
ABB = -ABB d P
In the same way the diamagnetic and paramagnetic contributions to the nuclear magnetic shielding tensor are derived: a$
= U$ + &
(18)
For the Landau transformation butions to the tensors are
(20) where
e(ri?- RI,) Iri - &I
(21)
=
x2 = ; (B,yz + B,zx + &xy)
x
P-re xx
PY XXY
-
e2 m2c2h. e
=
Wf”
+
AiB
Webb ~ WTB’ = wp” + Atl’” d
and the paramagnetic
w, MB’= w;”
wja
e2 m2c2h e
c ,fa wja
x [ ( ul~(YPzL~)
diamagnetic nuclear shielding contribution
(24) (25)
ones
W,BB+W~‘=W~+A; w+
_-
c J#a
(23)
the corresponding unitary operator is U = exp[-(ie/hc) Cy=i X(ri)]. Both second order energies (10) and (11) are left invariant for exact eigenfunctions, in the case of variational eigenfunctions belonging to a set of trial functions invariant to U [9]. This implies that a calculation of magnetic properties should fulfil constraints that can be expressed in the form of quantum mechanical sum rules. In a gauge transformation the diamagnetic contributions [lo] transform wf” -+ WdBB’
= 0, . . (29)
(other tensor components are obtained by cyclic permutation of the indices x, y and z). Off-diagonal diamagnetic components vanish for any coordinate system in the Landau gauge. The paramagnetic contribution is
(22)
induced by the generating function
[2], the contri-
X $;
is the electric field produced by the electron i on the position of the nucleus I. For the Landau transformation [2-41 of the Coulomb potential /I’ --+ A” = A’+VVX
(28)
X
Cl91
E;, =
71
364 (1996) 69-77
(26) + A?”
(27)
system, D’$” = 0;:” =
M.C. Caputo, M.B. Ferraro/Journal
12
cti” = 0. The paramagnetic contribution PIY = __ e2 0xX
m2c2h
of Molecular Structure (Theochem)
and other symbols have the same meaning as in [1,8], so that, for instance, the second order properties are
is
232
c
L X
V,,)-l =
[ (~,~~~141!t(~~~)i!U)]
etc., using the notation of previous papers [2]. The conditions for invariance = x$
(33)
0;;
= 0;;
(34)
imply, according to Eqs. (28), that the sum rules
etc. Sum rules for other components are obtained by cyclic permutations of the indices. We must also take into account the known sum rules for invariance of molecular magnetic properties through a translation of the origin of coordinates in the Coulomb gauge [12]
CL,, V,,)-i =~(~~~~Y~-z+) Vz,)-1 +
Fyr
Vyz)-1
VA,
= 2
CM;Y,
VyzJ-1
= -2
(35)
= 0
-
Pa,
(42) (43)
- &) = (JJ,, L&i w/3-1
=
(44)
~%$(4G,l4
These conditions, (35), (36) and (37), are equivalent to those found [2] directly from (33) and (34). The connection between diamagnetic contributions to susceptibility and shielding tensors in Coulomb and Landau gauges is easily found,
I)
( a!~xiGy!a)
(a!g
m,P$)(l
U
ZiEiz)
i=l
(I
(ML,
(p,, pp)-I = m,n&a
U e(ytEiy
(My,, Vy*)_* = 2
~Clse((al~,lj)(jlV,,la)) (3% ,fa wh
(41)
x$
&,
364 (1996) 69-77
(36)
viEb!a)
dQ xxx = $(x2
(45)
+ x;,“,
I dY XCKl= ZXcrcr dY
wyz7
VYZJ-l
(46)
=Qqu!fp+z:,!u)
(47) (48)
Cvyz9 vzx)-l
= T
(37)
( U!g(xy)i!a)
should be fulfilled. In these equations operator [9,1 l] has been introduced, VaO= i$(rOPfl
+PJp)i
= 2
b(O),
a virial
dI_V dIV um = uaa
(4%
These equations are satisfied exactly for any basis set. The relationships connecting paramagnetic contributions within the two gauges are [2]
g(rJ.)l]
XEF = xz xxyy= -
(38)
X !g
+ x:;
+ $ cx;,” - 3xZ)
xxyp-Lxg+x~~~x~y = x,“a
= x:p
= x$
(50) (51)
+x$
(52)
M.C. Caputo. M.B. Ferraro/Journal Table 1 Specification Basis Set
I II III
of Molecular Strucrure (Theochem)
364 (1996) 69- 77
13
of basis sets and SCF energies Contraction
Number CT0
scheme
GTO
CGTO
(14slOp4d/lOs6p4d) (13s10p2dlf/lls7p2d) (13slOp2dlf/l Is’lpld)
[Mp2d/5s3p2d] [6s5p2d 1f/6s5p2d] [6s5p2d I f/6s5pld]
GTO = Gaussian-type
Og? I%_
g.vy
=
ply_
orbitals,
CGTOS
= contracted
380 329 293
fl_Y>
-
plY 0J.Y
_ PI%? _ dI% - cry.7 O?‘.Y
(55)
= 0;; = &$ + iYdow
(56)
cT$YY+ ad,;jy = 0;;
+
c7$?
=
CT;;
(54)
Eqs. (50)-(55) would be exactly fulfilled only if the hypervirial theorem for the second moment operators (38) is obeyed, that is Ia) and lj) are “true” Hartree-Fock eigenstates [9].
3. Results and discussion In the present paper we report the diamagnetic and paramagnetic contributions to the magnetic susceptibility and magnetic shielding tensors in the SF6 molecule, evaluated via RPA [7] perturbation theory. We made the calculations for two choices of the gauge for the potential vector: the Coulomb and the Landau gauges [3,4]. We also included the treatment of general constraints for invariance of magnetic properties in a gauge transformation [l], and the sum rules relative to the Table 2 Sum rules’ (Eqs. (42) and (44)) for charge and current conservation and gauge invariance of magnetic properties (a.u.) for the SF6 molecule (non-vanishing contributions) Basis Set
W,,.,P,)-ib
(@)
(P<,>P,,)-I
n
I II III
1.528 2.821 2.772
-3.486 -3.515 -3.512
48.7 49.2 56.3
70 70 70
a Coordinates in bohr: S = (0, 0, O.O), F, = (0, 0, 2.9555314). F3 = (2.9555314, 0. 0). b (M;, y. P, )-I = -(M:,?.,
J’,)),
Number CGTOS
of
190 241 205
Gaussian-type
= Up,;%+; (CT;_;”- a:1.yV) (53) Ql% cTy?,
of
SCF energy (a.u.)
-994.20047968 -994.27943788 -994.32227784
orbitals.
transformation from the Coulomb to the Landau gauge are examined in detail. Lazzeretti et al. [ 131 published the ground state dipole polarizability and the diamagnetic shielding of the sulfur and fluorine nuclei of the SF6 molecule, calculated employing the coupled HartreeFock perturbation theory and exploiting the full Oh symmetry of the compound. We made our calculations employing the basis set proposed by Sadlej [14], Basis Set I, which is specially furnished to get better results of molecular properties with less computational effort. Our Basis Set III is that employed by Lazzeretti, containing 293 (205) primitive (contracted) functions: (13s10p2dlf/11s7pld)-[6s5p2d1f/6s5pld], for which the sulfur Gaussian basis set was taken from McLean and Chandler [15] and the fluorine basis set was taken from Van Duijneveldt [16]. The 3d functions on the S atom have exponents 1,5, 0.5 [17], those on F have exponent 1.0 [18] and the 4f set on S has exponent 1.O[ 171.To build Basis Set II, (13sl Op2d 1f/ 11s7p2d)-[6s5p2d 1f/6s5p2d], containing 329 (241) primitive (contracted) functions, we expanded the 3d set on the fluorine nucleus used by Lazzeretti, into two sets with exponents 2.47 and Table 3 Invariance condition (Eq. (35)) for the magnetic susceptibility of the SF6 molecule in the Landau gauge, evaluated with the origin on the fluorine nucleus, F, = (0, 0, 2.9555314) a.u., via Basis Sets I, II and III’ Basis Set
2@,., KY)_,”
(z2 - x2)
I II III
-7544.7 -7621.7 -8728.4
- 10853.7 - 10853.7 - 10853.7
a WY, Yr;)-l = -(k vr.-)-I. b Non-vanishing components.
M.C. Caputo, M.B. Ferraro/.lournal of Molecular Smcture
74
Table 4 Invariance condition (Eq. (37)) for the magnetic susceptibility of the SF, molecule in the Landau gauge, evaluated with the origin on the fluorine nucleus, Fi = (0, 0. 2.9555314) a.u., via Basis Sets I, II and IIIb Basis Set
4( I’,,,, VYl)_i”
(s’ +JS*)
4( Vz,%V.,)_,”
(z’ +x’)
I II III
15313.6 12724.0 15401.1
-20145.0 -20200.6 -20153.5
37947.7 35589.2 41586.0
-52706.6 -52761.8 -52714.0
b &on-vanishing
components.
0.07. In spite of the expansion of these polarization set of functions the SCF energy is larger than that corresponding to Basis Set III (see Table l), so we chose the order I, II, III to name the basis sets following the energy minimum. We made all the calculations, and determined the molecular properties and the sum rule constraints using the SYSMO program [ 10,121 and the extra RPA sections [ 1,2] implemented to describe molecular properties in the Landau gauge. Since we must use the highest Abelian subsymmetry to get the calculations in the RPA approximation, the DZh symmetry was used instead the full Oh symmetry of the molecule. So, we had to deal with disk files of more than 2 Gigabytes for Basis Sets II and III. The number of two-electron integrals larger than lo-” a.u. is 20078 904 for Basis Set I and 19 250 306 for Basis Set III. We used the experimental geometry (S-F distance = 1.564 A) quoted on [ 191. Table 2 reports the results for the invariance conditions for nuclear magnetic shielding (Eq. (44)). Table 5 Invariance ing of the origin on Basis Sets
condition (Eq. (36)) for the nuclear magnetic shieldSF6 molecule in the Landau gauge, evaluated with the the fluorine nucleus, F, = (0, 0, 2.9555314) a.u., via I, II and IIIb
Basis Set
2@4;,,.
I II III
251.8 235.5 277.0
V,,)_,”
(ZE; - IE,) 402.4 402.8 403.3
a (M;, I’ Vzr)-, = (MnF,,? V:,)-, = (M?,,, V,.,)-i = (M’;,,., V,,)-i = (MnF,), V,,)-, = (M?,;, VZV)_, = (M;,,, Vzl-)-, = 0 V:,)_i = -(M;,?., VzY)-i by symmetry. (M;,,, b Non-vanishing components.
( Theochem)
364 (1996)
69-77
The number of electrons provided through the Thomas-Reiche-Khun (42) sum rule, increases from 70 to 80% of the exact value (n = 70) when using Basis Sets I to III. These results are satisfactory but not very gratifying, because in spite of the quite large size of the basis sets, the wavefunction is not good enough to get a realistic description of the charge distribution. Numerical results relative to sum rules (35) to (37) are displayed in Tables 3-5. They demonstrate that basis sets of high quality are necessary to guarantee gauge invariance in a Landau transformation. Possibly, an even more severe probe for wavefunction accuracy is furnished by this constraint for magnetic susceptibility. To get a better understanding of the behavior of the magnetic susceptibility with the basis sets employed here, we include in Table 6 the results corresponding to two origins of the coordinate system: the center of mass (c.m.) and the fluorine nucleus (F,) for both gauges, Landau and Coulomb. The difference between the numerical results for these two origins is too large and constitutes an indication of the weakness of these basis sets in describing this property. Note also that the difference between x’ (c.m.) and xIp (c.m.) are of the same order as those between x’ (cm.) and xY VI). These features confirm our previous findings for small molecules [l], that the magnetic susceptibilities in the Landau gauge are quite a bit harder to evaluate to a fairly good degree of precision than Coulomb’s, even if the basis set is very large and flexible. This is easily understood, since total RPA magnetic susceptibilities are the sum of diamagnetic and paramagnetic contributions, (i.e. see Table 6, large numbers of different sign). These contributions become larger in absolute value in the Landau gauge, so that, although the accuracy of theoretical Landau contributions is comparable to Coulomb’s, the error is amplified by summing to get total susceptibilities. The results show that the quality of the basis sets employed is not good enough to evaluate properly the magnetic susceptibility. We may point out that calculations of the magnetic susceptibility might be used as an efficient test to verify if the wavefunction employed is in the “HF limit”. All the theoretical calculations of this
M.C. Caputo, M.B. Ferraro/Journal
of Molecular
Structure
Table 6 Magnetic susceptibility of SF6 in ppm a.u.a for the Coulomb (%?)band the Landau system, the center of mas (c.m.) and the nucleus F, = (0, 0, 2.9555314) Basis Set
XdV (cm.)
xpy(L,L)
I
-5036.4 -5050.2 -5038.4
3469.1 3187.4 3718.9
II III Exp.
(cm.)
(9)’
gauge evaluated
X dY (cm.)
X py
- 10072.7 -10100.0 -10076.8
7297.6 6368.5 1569. I
15
364 (1996) 69-77
(cm.)
at two origins of the coordinate
X’
(cm.)
-1567.3 - 1862.8 -1319.5
,yP
(cm.)
-2775.1 -3731.5 -2507.7
-493.06d -508.7’
I II III
xdV(4)
XP*(LsL)
-10463.2 -10477.0 -10465.2
7241.9 6998.3 8083.1
(4)
a The conversion factor from ppm a.u. per molecule b Magnetic susceptibilities in the Coulomb gauge. ’ Magnetic susceptibilities in the Landau gauge. d Taken from Refs. 120) and [Zl]. ’ Taken from Ref. [ZO].
xdY(Ft)
xpyVi)
xv(Ft1
xy(Fi)
-20926.5 -20954.0 -20930.5
14842.2 13990.2 16297.5
-3221 4 -3418.7 -2382.2
-6084.2 -6963.9 -4633.0
to the usual units of ppm cgs per mole is 8.9238878
Table 7 Magnetic susceptibility of SF6 in ppm a.u.a for the Coulomb of the coordinates at the center of mass Basis
(Theochem)
gauge using the formalisms
x lo-*.
derived from Geertsen’s
method
with the origin
dW
xp (‘% L)
xp W, L)
xd (KG)
xd (P, G)
xd (F, G)
X
3469.2 3187.4 3718.9
3542.0 2987.3 3557.5
-4104.3 -4149.4 -4561.6
-4045.1 -3391.3 -4075.6
-4764.3 -3433.0 -4113.1
-5036.4 -5050.2 -5038.4
x(W
xmd
x(FT
-635.1 -962.0 -842.7
-575.9 -203.9 -356.7 -493.06’ -S08.7g
-1222.3 -445.7 -555.6
Set
I II III
Total magnetic X I II III Exp.
susceptibilities
Vb
-1567.2 - 1862.7 -1319.5
a The conversion factor from ppm a.u. per molecule to the usual units of ppm cgs per mole is 8.923887810-2. b Magnetic susceptibilites in the Coulomb gauge from the same basis set. ’ Magnetic susceptibilites employing Geertsen’s method and the dipolar formalism for the diamagnetic x(R) = x(R, G)d + x(L, L)p from the same basis set. d Magnetic susceptibilites employing Geertsen’s method and the velocity formalism for the diamagnetic X(P) = X(P, G)d + x(L,L)~ from the same basis set. for the diamagnetic e Magnetic susceptibilies employing Geertsen’s method and the force formalism x(F) = x(F, G)d + x(K, L)P from the same basis set. f Taken from Refs. [20] and [21]. g Taken from Ref. [20].
contribution contribution contribution
M.C. Caputo, M.B. Ferraro/Journal
76
of Molecular Structure (Theochem)
property displayed in Table 6 are far from the experimental values found in the bibliography [20,21]. We also made the calculation of the diamagnetic contribution to the magnetic susceptibility, employing different formalisms, velocity (P), dipole length (R) and force (F), described in [6] and displayed in Table 7, in the RPA scheme proposed by Geertsen [5]. We found, for this “Polarization Propagator Based Theory”, that the total contributions for Basis Sets II and III are comparable to the experimental results [20,21]. Our previous conclusion [6] was that this formalism gives numerical susceptibilities comparable to accurate traditional Coulomb’s ones (Eqs. (16)-(17)) only in the “HF limit”. But in Table 8 Magnetic
shielding Nucleus
Component
0d@
up’g
I I1 III
S,’
Avg. Avg. Avg.
1363.7 1366.3 1366.4
-797.1 -887.5 -918.4
I
Fl
813.6 813.6 612.4 746.5
xx YY zz
Avg. II
Fl
xx YY zz
Avg. III
Fl
xx YY zi
Avg. Ex~.~ IIIe
F,
Ex~.~ IIP
F,
Fl
Fl Fi
this case, with a large number of electrons to deal with, Geertsen’s method seems to work better than the usual ones when the quality of the wavefunctions is poorer. It has been shown that the nuclear electric shielding in the angular momentum formalism is directly related to xP(K, L) [22], the paramagnetic contribution to the susceptibility in the mixed torqueangular momentum formalism, and also to the IR intensities [23]. So, those properties of electric origin might be badly reproduced without a better basis set. We think that a better basis set might be achieved if the exponents of the polarization set of functions of Basis III would be optimized searching for a maximum in xp, keeping in mind the idea of improving the results displayed in Table 6.
at sulfur Si and F,” (in ppm) in SF6 from Basis Sets I, II and IIIb
Basis
Sic SIC
364 (1996) 69-77
AU
crpy
c7v
ny
1363.7 1366.3 1366.4
-797.1 -887.5 -918.4
566.6 478.8 348.0
566.6 478.8 348.0
-626.3 -626.3 -197.7 -483.4
612.4 1014.8 612.4 746.5
-500.4 -752.2 -197.7 483.4
187.3 187.3 414.7 263.1
112.0 262.6 414.7 263.1
813.5 813.5 612.1 146.4
-658.0 -658.0 -179.7 -498.5
612.1 1014.9 612.1 746.4
-540.2 -775.7 -179.7 -498.5
155.5 155.5 432.4 247.9
71.9 239.2 432.4 247.9
813.7 813.7 612.1 746.5
-664.9 -664.9 -193.7 -507.8
612.1 1015.4 612.1 746.5
-526.3 -803.4 -193.7 -507.8
148.8 148.8 418.4 238.7
85.8 212.0 418.4 238.7
OdY
607 612.1
-256 ~ 193.7
351 418.4
811 813.7
-770 -664.9
41 148.3
310d
270.1e
a Coordinates in bohr: Si = (O,O, 0); Fl = (O,O, 2.9555314). b Origin of coordinates at the corresponding nucleus. ’ The diagonal components are identical to the average for Si because of symmetry. d Experimental values taken from Refs. 124,251. Parallel sign means parallel to the S-F e Values corresponding to Basis Set III in the Coulomb gauge.
bond.
M.C. Caputo. M.B. Ferraro/Journal
ofMolecularStruclute
Instead, the sum rule (36) corresponding to the invariance of nuclear magnetic shielding with the origin on the Fi nucleus, is better fulfilled, see Table 5, than those reported in Tables 3 and 4. The nuclear magnetic shielding displayed in Table 8 confirm that the basis sets employed here are good enough to describe them owing to the similarity between the values obtained for the two gauges. Coulomb and Landau. Even larger deviations from symmetry requirements occur for a:; and 0:; for the Landau gauge since the symmetry constraint YF, _ flX.X - g_y1 is poorly obeyed by the three basis sets, but the average cfVI: = 0::;. is verified. We include in Table 8 the fluorine experimental shieldings reported by Garg et al. [24]. They found a chemical shift anisotropy, Au, of nearly 300 ppm, and determined the bond length S-F to be 1.58 f 0.01 A by NMR methods. From the value of 0~“. [25], they estimated the individual components cl1 and cI (parallel sign means parallel to the S-F bond). The corresponding diamagnetic contributions were evaluated semiempirically and the paramagnetic ones, by difference between total and diamagnetic contributions. Our ab initio results of shielding contributions for Basis Sets II and III are acceptable if we consider that we made the calculations employing the experimental bond length of 1.564 A [19]. Tables 2 and 5, corresponding to gauge invariance constraints to be fulfilled in magnetic shielding calculations, show that Basis Sets II and III are good enough to represent the magnetic shielding on SF6. The behavior observed in the calculation shown here, makes us consider the SF, compound to be a very good example to get a deeper insight into the gauge invariance constraints, and the fulfillment of the set of sum rules [8] provided by the virial theorem [4,9].
Acknowledgments
Financial support for the present research from
(Theochem)
the University acknowledged.
364 (1996) 69-77
of Buenos
Aires
71
is gratefully
References [ll P. Lazzeretti, M. Malagoli, R. Zanasi, M.B. Ferraro and M.C. Caputo, J. Chem. Phys., 103 (1995) 1852. PI M.B. Ferraro, T.E. Herr, P. Lazzeretti, M. Malagoli and R. Zanasi, Phys. Rev. A, 45 (1992) 6272; J. Chem. Phys., 98 (1993) 4030. [31 L. Landau and E. Lifshitz, Theorie de Champs, 3rd edn., Mir, Moscow, 1970. Quantique, Mir, [41 L. Landau and E. Lifshitz, Micanique Moscow, 1966. [51 J. Geertsen, J. Chem. Phys., 90 (1989) 4892; Chem. Phys. Lett., 179, (1991) 479; 188 (1992) 326. 161 M.B. Ferraro and M.C. Caputo, J. Mol. Struct. (Theothem), 335 (1995) 69. 171 D.J. Rowe, Rev. Mod. Phys., 40 (1968) 153. 181 P. Lazzeretti, Adv. Chem. Phys., 75 (1987) 507. 191 ST. Epstein, The Variation Method in Quantum Chemistry, Academic Press, New York, 1974. HOI P. Lazzeretti and R. Zanasi, Phys. Rev. A, 32 (1985) 2607. 1111 P. Lazzeretti, Theor. Chim. Acta, 87 (1993) 59. [121 P. Lazzeretti and R. Zanasi, Phys. Rev. A, 33 (1986) 3727. 1131 P. Lazzeretti, U. Pincelli, E. Rossi and R. Zanasi, J. Chem. Phys., 79 (1983) 4085. [141 A.J. Sadlej, Collect. Czech Chem. Commun., 53 (1988) 1995. J. Chem. Phys., 72 [151 A.D. McLean and G.S. Chandler, (1980) 5639; IBM Research Report RJ2665 (34180) 1979. [161 F.B. Van Duijneveldt, IBM Research Report RJ945, 1971. 1171 W. von Niessen, W.P. Kraemer and G.H.F. Diercksen, Chem. Phys. Lett., 63 (1979) 65. Van Duijneveldt-Van de Rijdt and F.B. Van 1181 J.G.C.M. Duijneveldt, J. Mol. Struct. (Theochem), 89 (1982) 185. [191 G. Herzberg, Electronic Spectra and Electronic Structures of Polyatomic Molecules, Van Nostrand, Princeton, NJ, 1966. [201 P. Henkel and W. Klemm, Z. Anorg. Allg. Chem., 222 (1935) 70. of Chemistry and Physics, 65th edn., CRC [211 Handbook Press, Boca Raton, FL, 1984/1985. [221 P. Lazzeretti, R. Zanasi, T. Prosperi and A. Lapiccirella, Chem. Phys. Lett., 150 (1988) 115. 1231 P. Lazzeretti and R. Zanasi, Chem. Phys. Lett., 112 (1984) 13. 1241S.K. Garg, J.A. Ripmeester and D.W. Davidson, J. Magn. Reson., 39 (1980) 317. P51 S. Mohanty, J. Chem. Phys., 59 (1973) 4415.
Description of the gauge invariance of molecular magnetic properties of the SF6 molecule M.C. Caputo, M.B. Ferraro*” Departamento
de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria. Buenos Aires, Argentina
Pab. I, (1428)
Received 17 August 1995; accepted 16 November 1995
Abstract Ab initio calculations of the magnetic susceptibility and the nuclear magnetic shielding tensor for the SF, molecule are reported here. Their dependence on the choice of gauge for the vector potential, which describes the external magnetic field, is discussed. The required invariance of molecular magnetic properties depends on the fulfillment of very general gauge-invariance sum rules and on the quality of the basis set employed. Keywords;
Gauge
invariance; Magnetic shielding tensor; Magnetic susceptibility
1. Introduction
The magnetic susceptibility and the nuclear magnetic shielding tensor are molecular properties for which the gauge invariance of the calculated quantities is a physical requirement. A translational transformation of the origin of coordinates may also be interpreted as a gauge transformation of the vector potential. In the past, we have analyzed [l] gauge constraints for different choices of those gauge transformations. For instance, we analyzed exhaustively the Landau transformation [2-41 and the polarization propagator based theory proposed by Geertsen [5] for a series of small molecules: CH4, NH3, HF and HZ0 [6].
* Corresponding author. ’ Member of Carrera de1 Investigador 0166-1280/96/$15.00 SSDZ
de1 CONICET.
In this case we have studied the molecular magnetic properties of the SF6 molecule, which is larger than those analyzed previously by us; the number of electrons to deal with is 70. We evaluated these properties for the Coulomb and the Landau gauges, employing extended Gaussian basis sets to represent the wavefunction, and the Random Phase Approximation (RPA) [7] to describe the perturbation scheme. For the magnetic susceptibility, we also made calculations for two origins of the coordinates and for the different formalisms described in [6] for the diamagnetic susceptibility. The degree to which very general sum rules are fulfilled is a measure of the quality of the basis set employed for determining theoretical molecular properties. General constraints for invariance of magnetic properties in an arbitrary gauge transformation have been analyzed [l] and those corresponding to the transformation from
0 1996 Elsevier Science B.V. All rights reserved
0166-1280(95)04455-8
M.C. Capulo, M.B. Ferraro/Journal
70
of Molecular Structure (Theochem)
Coulomb to Landau gauges were examined in particular. In Section 2 the expressions for the magnetic susceptibility and the nuclear magnetic shielding tensor in the Coulomb and the Landau gauges are given. The sum rules for invariance under a Landau transformation are also included. Section 3 discusses the numerical results for the SF, molecule.
364 (1996) 69-77
been introduced, i.e.,
e2 Ay* HBB =--m,c2 ’
A: = A’(Q)
(7)
If a permanent intrinsic magnetic dipole moment ,u,, placed on nucleus I, with vector potential A’“‘(r) = -pLI x VP Ir -‘&I
2. Landau transformation and its conditions for invariance of magnetic susceptibility and nuclear magnetic shielding tensor
is taken into account, the interaction Hamiltonian contains the additional terms
Let us consider a molecule with ni electrons, with mass m,, charge -e, coordinates ri, canonical moments pi, angular momenta ri = ri x pi (i = 1, 2 . . n), and N nuclei, with corresponding quantitiks ZIe, M,, RI, etc. The “particle” Hamiltonian of the electrons is
e2 e -~(A~‘.p)i+~~(A’.A”)i met i=, m,c2 i=*
(9)
The second order energies wBB = -&+B,B~
= w,“” + wp
‘B p = wt4,B+ wp W MB -_ &a~,/3
(10) (11)
are defined for the Coulomb gauge by
(12) 1 N N Z1ZJe2 +jcc I Jzl I& - RJI
(1)
with eigenstates ]j) and energy eigenvalues E(O); the reference state is denoted by ]u) (the n$ation of related papers [1,8] is retained here, e.g. L, = Cyzl li,; wja = i(Ey’ - Ei’)), etc.). In the presence of a uniform magnetic field B, the divergenceless Coulomb vector potential is a transverse field, defined by the equation A’(r)
= i/l x r
V-A’=0
P
e2 h? c w. 2m2c2A e ~#a Ja
(14)
(2)
(3)
The total Hamiltonian is represented by a truncated Taylor series, H=H”+HB+$HEB
wBB=-
(4)
where the interaction Hamiltonian, containing terms of first and second order in the field, has
(15) In the reference state ]u) the diamagnetic contribution to the magnetic susceptibility tensor is
(16)
M.C. Caputo, M.B. Ferraro/Journal
and the paramagnetic contribution
of Molecular Structure (Theochem)
Therefore, under such gauge transformation
is
ApPIB _ --AT”
ABB = -ABB d P
In the same way the diamagnetic and paramagnetic contributions to the nuclear magnetic shielding tensor are derived: a$
= U$ + &
(18)
For the Landau transformation butions to the tensors are
(20) where
e(ri?- RI,) Iri - &I
(21)
=
x2 = ; (B,yz + B,zx + &xy)
x
P-re xx
PY XXY
-
e2 m2c2h. e
=
Wf”
+
AiB
Webb ~ WTB’ = wp” + Atl’” d
and the paramagnetic
w, MB’= w;”
wja
e2 m2c2h e
c ,fa wja
x [ ( ul~(YPzL~)
diamagnetic nuclear shielding contribution
(24) (25)
ones
W,BB+W~‘=W~+A; w+
_-
c J#a
(23)
the corresponding unitary operator is U = exp[-(ie/hc) Cy=i X(ri)]. Both second order energies (10) and (11) are left invariant for exact eigenfunctions, in the case of variational eigenfunctions belonging to a set of trial functions invariant to U [9]. This implies that a calculation of magnetic properties should fulfil constraints that can be expressed in the form of quantum mechanical sum rules. In a gauge transformation the diamagnetic contributions [lo] transform wf” -+ WdBB’
= 0, . . (29)
(other tensor components are obtained by cyclic permutation of the indices x, y and z). Off-diagonal diamagnetic components vanish for any coordinate system in the Landau gauge. The paramagnetic contribution is
(22)
induced by the generating function
[2], the contri-
X $;
is the electric field produced by the electron i on the position of the nucleus I. For the Landau transformation [2-41 of the Coulomb potential /I’ --+ A” = A’+VVX
(28)
X
Cl91
E;, =
71
364 (1996) 69-77
(26) + A?”
(27)
system, D’$” = 0;:” =
M.C. Caputo, M.B. Ferraro/Journal
12
cti” = 0. The paramagnetic contribution PIY = __ e2 0xX
m2c2h
of Molecular Structure (Theochem)
and other symbols have the same meaning as in [1,8], so that, for instance, the second order properties are
is
232
c
L X
V,,)-l =
[ (~,~~~141!t(~~~)i!U)]
etc., using the notation of previous papers [2]. The conditions for invariance = x$
(33)
0;;
= 0;;
(34)
imply, according to Eqs. (28), that the sum rules
etc. Sum rules for other components are obtained by cyclic permutations of the indices. We must also take into account the known sum rules for invariance of molecular magnetic properties through a translation of the origin of coordinates in the Coulomb gauge [12]
CL,, V,,)-i =~(~~~~Y~-z+) Vz,)-1 +
Fyr
Vyz)-1
VA,
= 2
CM;Y,
VyzJ-1
= -2
(35)
= 0
-
Pa,
(42) (43)
- &) = (JJ,, L&i w/3-1
=
(44)
~%$(4G,l4
These conditions, (35), (36) and (37), are equivalent to those found [2] directly from (33) and (34). The connection between diamagnetic contributions to susceptibility and shielding tensors in Coulomb and Landau gauges is easily found,
I)
( a!~xiGy!a)
(a!g
m,P$)(l
U
ZiEiz)
i=l
(I
(ML,
(p,, pp)-I = m,n&a
U e(ytEiy
(My,, Vy*)_* = 2
~Clse((al~,lj)(jlV,,la)) (3% ,fa wh
(41)
x$
&,
364 (1996) 69-77
(36)
viEb!a)
dQ xxx = $(x2
(45)
+ x;,“,
I dY XCKl= ZXcrcr dY
wyz7
VYZJ-l
(46)
=Qqu!fp+z:,!u)
(47) (48)
Cvyz9 vzx)-l
= T
(37)
( U!g(xy)i!a)
should be fulfilled. In these equations operator [9,1 l] has been introduced, VaO= i$(rOPfl
+PJp)i
= 2
b(O),
a virial
dI_V dIV um = uaa
(4%
These equations are satisfied exactly for any basis set. The relationships connecting paramagnetic contributions within the two gauges are [2]
g(rJ.)l]
XEF = xz xxyy= -
(38)
X !g
+ x:;
+ $ cx;,” - 3xZ)
xxyp-Lxg+x~~~x~y = x,“a
= x:p
= x$
(50) (51)
+x$
(52)
M.C. Caputo. M.B. Ferraro/Journal Table 1 Specification Basis Set
I II III
of Molecular Strucrure (Theochem)
364 (1996) 69- 77
13
of basis sets and SCF energies Contraction
Number CT0
scheme
GTO
CGTO
(14slOp4d/lOs6p4d) (13s10p2dlf/lls7p2d) (13slOp2dlf/l Is’lpld)
[Mp2d/5s3p2d] [6s5p2d 1f/6s5p2d] [6s5p2d I f/6s5pld]
GTO = Gaussian-type
Og? I%_
g.vy
=
ply_
orbitals,
CGTOS
= contracted
380 329 293
fl_Y>
-
plY 0J.Y
_ PI%? _ dI% - cry.7 O?‘.Y
(55)
= 0;; = &$ + iYdow
(56)
cT$YY+ ad,;jy = 0;;
+
c7$?
=
CT;;
(54)
Eqs. (50)-(55) would be exactly fulfilled only if the hypervirial theorem for the second moment operators (38) is obeyed, that is Ia) and lj) are “true” Hartree-Fock eigenstates [9].
3. Results and discussion In the present paper we report the diamagnetic and paramagnetic contributions to the magnetic susceptibility and magnetic shielding tensors in the SF6 molecule, evaluated via RPA [7] perturbation theory. We made the calculations for two choices of the gauge for the potential vector: the Coulomb and the Landau gauges [3,4]. We also included the treatment of general constraints for invariance of magnetic properties in a gauge transformation [l], and the sum rules relative to the Table 2 Sum rules’ (Eqs. (42) and (44)) for charge and current conservation and gauge invariance of magnetic properties (a.u.) for the SF6 molecule (non-vanishing contributions) Basis Set
W,,.,P,)-ib
(@)
(P<,>P,,)-I
n
I II III
1.528 2.821 2.772
-3.486 -3.515 -3.512
48.7 49.2 56.3
70 70 70
a Coordinates in bohr: S = (0, 0, O.O), F, = (0, 0, 2.9555314). F3 = (2.9555314, 0. 0). b (M;, y. P, )-I = -(M:,?.,
J’,)),
Number CGTOS
of
190 241 205
Gaussian-type
= Up,;%+; (CT;_;”- a:1.yV) (53) Ql% cTy?,
of
SCF energy (a.u.)
-994.20047968 -994.27943788 -994.32227784
orbitals.
transformation from the Coulomb to the Landau gauge are examined in detail. Lazzeretti et al. [ 131 published the ground state dipole polarizability and the diamagnetic shielding of the sulfur and fluorine nuclei of the SF6 molecule, calculated employing the coupled HartreeFock perturbation theory and exploiting the full Oh symmetry of the compound. We made our calculations employing the basis set proposed by Sadlej [14], Basis Set I, which is specially furnished to get better results of molecular properties with less computational effort. Our Basis Set III is that employed by Lazzeretti, containing 293 (205) primitive (contracted) functions: (13s10p2dlf/11s7pld)-[6s5p2d1f/6s5pld], for which the sulfur Gaussian basis set was taken from McLean and Chandler [15] and the fluorine basis set was taken from Van Duijneveldt [16]. The 3d functions on the S atom have exponents 1,5, 0.5 [17], those on F have exponent 1.0 [18] and the 4f set on S has exponent 1.O[ 171.To build Basis Set II, (13sl Op2d 1f/ 11s7p2d)-[6s5p2d 1f/6s5p2d], containing 329 (241) primitive (contracted) functions, we expanded the 3d set on the fluorine nucleus used by Lazzeretti, into two sets with exponents 2.47 and Table 3 Invariance condition (Eq. (35)) for the magnetic susceptibility of the SF6 molecule in the Landau gauge, evaluated with the origin on the fluorine nucleus, F, = (0, 0, 2.9555314) a.u., via Basis Sets I, II and III’ Basis Set
2@,., KY)_,”
(z2 - x2)
I II III
-7544.7 -7621.7 -8728.4
- 10853.7 - 10853.7 - 10853.7
a WY, Yr;)-l = -(k vr.-)-I. b Non-vanishing components.
M.C. Caputo, M.B. Ferraro/.lournal of Molecular Smcture
74
Table 4 Invariance condition (Eq. (37)) for the magnetic susceptibility of the SF, molecule in the Landau gauge, evaluated with the origin on the fluorine nucleus, Fi = (0, 0. 2.9555314) a.u., via Basis Sets I, II and IIIb Basis Set
4( I’,,,, VYl)_i”
(s’ +JS*)
4( Vz,%V.,)_,”
(z’ +x’)
I II III
15313.6 12724.0 15401.1
-20145.0 -20200.6 -20153.5
37947.7 35589.2 41586.0
-52706.6 -52761.8 -52714.0
b &on-vanishing
components.
0.07. In spite of the expansion of these polarization set of functions the SCF energy is larger than that corresponding to Basis Set III (see Table l), so we chose the order I, II, III to name the basis sets following the energy minimum. We made all the calculations, and determined the molecular properties and the sum rule constraints using the SYSMO program [ 10,121 and the extra RPA sections [ 1,2] implemented to describe molecular properties in the Landau gauge. Since we must use the highest Abelian subsymmetry to get the calculations in the RPA approximation, the DZh symmetry was used instead the full Oh symmetry of the molecule. So, we had to deal with disk files of more than 2 Gigabytes for Basis Sets II and III. The number of two-electron integrals larger than lo-” a.u. is 20078 904 for Basis Set I and 19 250 306 for Basis Set III. We used the experimental geometry (S-F distance = 1.564 A) quoted on [ 191. Table 2 reports the results for the invariance conditions for nuclear magnetic shielding (Eq. (44)). Table 5 Invariance ing of the origin on Basis Sets
condition (Eq. (36)) for the nuclear magnetic shieldSF6 molecule in the Landau gauge, evaluated with the the fluorine nucleus, F, = (0, 0, 2.9555314) a.u., via I, II and IIIb
Basis Set
2@4;,,.
I II III
251.8 235.5 277.0
V,,)_,”
(ZE; - IE,) 402.4 402.8 403.3
a (M;, I’ Vzr)-, = (MnF,,? V:,)-, = (M?,,, V,.,)-i = (M’;,,., V,,)-i = (MnF,), V,,)-, = (M?,;, VZV)_, = (M;,,, Vzl-)-, = 0 V:,)_i = -(M;,?., VzY)-i by symmetry. (M;,,, b Non-vanishing components.
( Theochem)
364 (1996)
69-77
The number of electrons provided through the Thomas-Reiche-Khun (42) sum rule, increases from 70 to 80% of the exact value (n = 70) when using Basis Sets I to III. These results are satisfactory but not very gratifying, because in spite of the quite large size of the basis sets, the wavefunction is not good enough to get a realistic description of the charge distribution. Numerical results relative to sum rules (35) to (37) are displayed in Tables 3-5. They demonstrate that basis sets of high quality are necessary to guarantee gauge invariance in a Landau transformation. Possibly, an even more severe probe for wavefunction accuracy is furnished by this constraint for magnetic susceptibility. To get a better understanding of the behavior of the magnetic susceptibility with the basis sets employed here, we include in Table 6 the results corresponding to two origins of the coordinate system: the center of mass (c.m.) and the fluorine nucleus (F,) for both gauges, Landau and Coulomb. The difference between the numerical results for these two origins is too large and constitutes an indication of the weakness of these basis sets in describing this property. Note also that the difference between x’ (c.m.) and xIp (c.m.) are of the same order as those between x’ (cm.) and xY VI). These features confirm our previous findings for small molecules [l], that the magnetic susceptibilities in the Landau gauge are quite a bit harder to evaluate to a fairly good degree of precision than Coulomb’s, even if the basis set is very large and flexible. This is easily understood, since total RPA magnetic susceptibilities are the sum of diamagnetic and paramagnetic contributions, (i.e. see Table 6, large numbers of different sign). These contributions become larger in absolute value in the Landau gauge, so that, although the accuracy of theoretical Landau contributions is comparable to Coulomb’s, the error is amplified by summing to get total susceptibilities. The results show that the quality of the basis sets employed is not good enough to evaluate properly the magnetic susceptibility. We may point out that calculations of the magnetic susceptibility might be used as an efficient test to verify if the wavefunction employed is in the “HF limit”. All the theoretical calculations of this
M.C. Caputo, M.B. Ferraro/Journal
of Molecular
Structure
Table 6 Magnetic susceptibility of SF6 in ppm a.u.a for the Coulomb (%?)band the Landau system, the center of mas (c.m.) and the nucleus F, = (0, 0, 2.9555314) Basis Set
XdV (cm.)
xpy(L,L)
I
-5036.4 -5050.2 -5038.4
3469.1 3187.4 3718.9
II III Exp.
(cm.)
(9)’
gauge evaluated
X dY (cm.)
X py
- 10072.7 -10100.0 -10076.8
7297.6 6368.5 1569. I
15
364 (1996) 69-77
(cm.)
at two origins of the coordinate
X’
(cm.)
-1567.3 - 1862.8 -1319.5
,yP
(cm.)
-2775.1 -3731.5 -2507.7
-493.06d -508.7’
I II III
xdV(4)
XP*(LsL)
-10463.2 -10477.0 -10465.2
7241.9 6998.3 8083.1
(4)
a The conversion factor from ppm a.u. per molecule b Magnetic susceptibilities in the Coulomb gauge. ’ Magnetic susceptibilities in the Landau gauge. d Taken from Refs. 120) and [Zl]. ’ Taken from Ref. [ZO].
xdY(Ft)
xpyVi)
xv(Ft1
xy(Fi)
-20926.5 -20954.0 -20930.5
14842.2 13990.2 16297.5
-3221 4 -3418.7 -2382.2
-6084.2 -6963.9 -4633.0
to the usual units of ppm cgs per mole is 8.9238878
Table 7 Magnetic susceptibility of SF6 in ppm a.u.a for the Coulomb of the coordinates at the center of mass Basis
(Theochem)
gauge using the formalisms
x lo-*.
derived from Geertsen’s
method
with the origin
dW
xp (‘% L)
xp W, L)
xd (KG)
xd (P, G)
xd (F, G)
X
3469.2 3187.4 3718.9
3542.0 2987.3 3557.5
-4104.3 -4149.4 -4561.6
-4045.1 -3391.3 -4075.6
-4764.3 -3433.0 -4113.1
-5036.4 -5050.2 -5038.4
x(W
xmd
x(FT
-635.1 -962.0 -842.7
-575.9 -203.9 -356.7 -493.06’ -S08.7g
-1222.3 -445.7 -555.6
Set
I II III
Total magnetic X I II III Exp.
susceptibilities
Vb
-1567.2 - 1862.7 -1319.5
a The conversion factor from ppm a.u. per molecule to the usual units of ppm cgs per mole is 8.923887810-2. b Magnetic susceptibilites in the Coulomb gauge from the same basis set. ’ Magnetic susceptibilites employing Geertsen’s method and the dipolar formalism for the diamagnetic x(R) = x(R, G)d + x(L, L)p from the same basis set. d Magnetic susceptibilites employing Geertsen’s method and the velocity formalism for the diamagnetic X(P) = X(P, G)d + x(L,L)~ from the same basis set. for the diamagnetic e Magnetic susceptibilies employing Geertsen’s method and the force formalism x(F) = x(F, G)d + x(K, L)P from the same basis set. f Taken from Refs. [20] and [21]. g Taken from Ref. [20].
contribution contribution contribution
M.C. Caputo, M.B. Ferraro/Journal
76
of Molecular Structure (Theochem)
property displayed in Table 6 are far from the experimental values found in the bibliography [20,21]. We also made the calculation of the diamagnetic contribution to the magnetic susceptibility, employing different formalisms, velocity (P), dipole length (R) and force (F), described in [6] and displayed in Table 7, in the RPA scheme proposed by Geertsen [5]. We found, for this “Polarization Propagator Based Theory”, that the total contributions for Basis Sets II and III are comparable to the experimental results [20,21]. Our previous conclusion [6] was that this formalism gives numerical susceptibilities comparable to accurate traditional Coulomb’s ones (Eqs. (16)-(17)) only in the “HF limit”. But in Table 8 Magnetic
shielding Nucleus
Component
0d@
up’g
I I1 III
S,’
Avg. Avg. Avg.
1363.7 1366.3 1366.4
-797.1 -887.5 -918.4
I
Fl
813.6 813.6 612.4 746.5
xx YY zz
Avg. II
Fl
xx YY zz
Avg. III
Fl
xx YY zi
Avg. Ex~.~ IIIe
F,
Ex~.~ IIP
F,
Fl
Fl Fi
this case, with a large number of electrons to deal with, Geertsen’s method seems to work better than the usual ones when the quality of the wavefunctions is poorer. It has been shown that the nuclear electric shielding in the angular momentum formalism is directly related to xP(K, L) [22], the paramagnetic contribution to the susceptibility in the mixed torqueangular momentum formalism, and also to the IR intensities [23]. So, those properties of electric origin might be badly reproduced without a better basis set. We think that a better basis set might be achieved if the exponents of the polarization set of functions of Basis III would be optimized searching for a maximum in xp, keeping in mind the idea of improving the results displayed in Table 6.
at sulfur Si and F,” (in ppm) in SF6 from Basis Sets I, II and IIIb
Basis
Sic SIC
364 (1996) 69-77
AU
crpy
c7v
ny
1363.7 1366.3 1366.4
-797.1 -887.5 -918.4
566.6 478.8 348.0
566.6 478.8 348.0
-626.3 -626.3 -197.7 -483.4
612.4 1014.8 612.4 746.5
-500.4 -752.2 -197.7 483.4
187.3 187.3 414.7 263.1
112.0 262.6 414.7 263.1
813.5 813.5 612.1 146.4
-658.0 -658.0 -179.7 -498.5
612.1 1014.9 612.1 746.4
-540.2 -775.7 -179.7 -498.5
155.5 155.5 432.4 247.9
71.9 239.2 432.4 247.9
813.7 813.7 612.1 746.5
-664.9 -664.9 -193.7 -507.8
612.1 1015.4 612.1 746.5
-526.3 -803.4 -193.7 -507.8
148.8 148.8 418.4 238.7
85.8 212.0 418.4 238.7
OdY
607 612.1
-256 ~ 193.7
351 418.4
811 813.7
-770 -664.9
41 148.3
310d
270.1e
a Coordinates in bohr: Si = (O,O, 0); Fl = (O,O, 2.9555314). b Origin of coordinates at the corresponding nucleus. ’ The diagonal components are identical to the average for Si because of symmetry. d Experimental values taken from Refs. 124,251. Parallel sign means parallel to the S-F e Values corresponding to Basis Set III in the Coulomb gauge.
bond.
M.C. Caputo. M.B. Ferraro/Journal
ofMolecularStruclute
Instead, the sum rule (36) corresponding to the invariance of nuclear magnetic shielding with the origin on the Fi nucleus, is better fulfilled, see Table 5, than those reported in Tables 3 and 4. The nuclear magnetic shielding displayed in Table 8 confirm that the basis sets employed here are good enough to describe them owing to the similarity between the values obtained for the two gauges. Coulomb and Landau. Even larger deviations from symmetry requirements occur for a:; and 0:; for the Landau gauge since the symmetry constraint YF, _ flX.X - g_y1 is poorly obeyed by the three basis sets, but the average cfVI: = 0::;. is verified. We include in Table 8 the fluorine experimental shieldings reported by Garg et al. [24]. They found a chemical shift anisotropy, Au, of nearly 300 ppm, and determined the bond length S-F to be 1.58 f 0.01 A by NMR methods. From the value of 0~“. [25], they estimated the individual components cl1 and cI (parallel sign means parallel to the S-F bond). The corresponding diamagnetic contributions were evaluated semiempirically and the paramagnetic ones, by difference between total and diamagnetic contributions. Our ab initio results of shielding contributions for Basis Sets II and III are acceptable if we consider that we made the calculations employing the experimental bond length of 1.564 A [19]. Tables 2 and 5, corresponding to gauge invariance constraints to be fulfilled in magnetic shielding calculations, show that Basis Sets II and III are good enough to represent the magnetic shielding on SF6. The behavior observed in the calculation shown here, makes us consider the SF, compound to be a very good example to get a deeper insight into the gauge invariance constraints, and the fulfillment of the set of sum rules [8] provided by the virial theorem [4,9].
Acknowledgments
Financial support for the present research from
(Theochem)
the University acknowledged.
364 (1996) 69-77
of Buenos
Aires
71
is gratefully
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