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Some calculations with magnetic field dependent orbitals
Volume 57, number 3
SOME CALCULATIONS
M_ ZAI~~ER*, Brernical Instihtre
WITH MAGNETIC
D_ PUMPERS Boris Kid&,
1 August 1978
PHYSICS LETrE.E.
CHEMICAL
FiELD
DEPENDENT
ORBITALS
and A. A&MN Ljubljana, Yugoshv~
Received 13 ApriI 1978
Calculatious with magnetic field dependent orbit&, which include usual cartes& gaus.Gans md London’s tieid dependent orbit& as speciaI cases, are presentzxi_Magnetic suscep?lbiiity and magnetic shielding of Hz and HF moIecuIer are calculated within the fiite perturbation Hartree-Fock LCAO method.
1. introduction A theoretical description of molecular magnetic susceptibilities and shielding constants has been a difficult problem mainly because of the gauge invariance of the vector potential of the external magnetic field [l] . Finite basis set calculations show a large dependence on the origin of the vector potential [2] and even for elaborate basis sets the computed second order molecular magnetic properties reveal a considerable gauge origin dependence [3]. As a way of avoiding this difficulty London [4] suggested the use of a‘particular form of field dependent orbit& for the basis set (nocalkd gauge invariant atomic orbitals - GIAO’s). Since then London orbitals have been widely used and the discussion about their physical meaning and applicability in calculations of magnetic properties is going on_ Though such a basis set forces the gauge invariance of the energy, a problem with current conservation appeared [5,6] _ Epstein also pointed out [S] that the constant vector potential in the phase factor could be effectively varied to preserve the translational invariance and to remove some non-physical properties of current. Despite this problem London orbitals are considered to have the advantage of being optimal and correct through first order in magnetic field density for the one-center problem- Using these orbitals good results f7,8] have been obtained for magnetic shieklings and susceptibilities. Some authors are however inclined to the idea that the use of magnetic field dependent orbit& leads to the appropriate extension of the virtual spectra [9]. A very recent application of the field dependent orbitals is by Sadlej [lo] who did calculations of second order electric properties ofmolecules with considerable ,zsccess. The purpose of this paper is to present calculations of magnetic susceptibility and shiel ‘ing in the frame of the field dependent orbitals, which include a variational paramerer p in the phase factor. For & ecial values of these factors they redze to London orbitals @ = 1) and to the gaussian basis set @ = 0). One result of this calculation is that in some cases London orbit& give a worse susceptiiility than the field independent gaussian type orbitals (GTWs)_
2. Theory The -kinetic part of the one-electron
-totian
* Presentaddress: Term&a. Ljtiljana, Y~gosIavia. 338
for the system
in an external
magnetic field is
CHEMICAL PKYSICS LErlERs
VoIume 57. num’oer3
HI = -(fi2f2nz)V2
- (iefi/rn)A.V
1 August 1978
f (e2/2m)A2_
(1)
The vector potential is taken to be A(r) = $B x (r - RG). where RG is defmed ro some arbitrary origin. The magnetic field dependent orbital centered at nuclear site C is @c = x&$$
exp (-o$)
exp C-(iep/WC
-r] -
!a
The matrix elements of H, are given zs a sum of three terms: <@&Zr@c> = -(ti2/2m)KDc
+ (iefi/2m)L,c
f (e2/h)QW,
(3)
where (omitting indices D and C) K = 402 [S(Zi2) + S(m+2) + S(n+2)]
- a[4(Z+m+n) + 61 S
+ Z(Z- 1) S(Z-2) f m(??z- 1) S(m-2) L =BJ?rrS(??z-l,n+l)
+m(l-p)ZccS(m-l)
-n(l-p)YccS(n-1)
+n(l-p)XCGS(“-l)
S(Z-i-1) - Xcc
+Z(l-p)YoS(Z-l)
--m(l-_p)XcGS(m-l)
-Z,,S(m+l)]] - Ls(Z-l,n+l)
S(Z- 1) + 24 l--p) 12,
+B,{ZS(Z-l,m+l)
(4)
-RzS(?n+l,n-1)
+ 2&&-p)[Y~&?2+1)
+By{nS(Z+l,n-l) - Z(l-p)Zcc
+ n(?z- 1) S(ll-2),
S(n+l)]
)
-mS(Z+l.m-1)
+20(l-P)[XcGS(m+l)
- Y,S(Z+-l)]],
Q=Z3;{S(m+2)+S(n+2)+2(1--p)[YccS(m+l)+&S(n+l)] +B;
{S(n+2)
+S(z+2) f 2(1-p)[&
+ B;
{S(Z+2) + S(m+2)
- 2BxBz
+(I-~)~(Y&+zzy;)S}
S(?zH) +xcc
+ (I-P)[Z&
S(Z+l) +Xcc
S(m+l)]
S(Z+l) + XcG S(n+l)]
+fl--p)[&$m+l)
In these expressions S is the overlap integral and X,,
+Qs)
S(Z+1)] + ( 1-p)2(z&
+ 2( 1 -p) [XCc S(Z+l) i- Ycc S(m+l)]
- 2B,B,, @(I+1 ,m+l) + ( l-&[YcG
(5)
f ( l-~)~
(X&
+ (‘-P)~X~&~ + ( 1-p)2X&cc
+X&(n+l)]
(or YcG, Zcc)
+(:-p)2YCCZCG
+ Y&)
S3
S] S] S)
.
(6)
is defmed as
All parameters (o, Z,m, rz, R,, p) that appear in eqs. (4)-(6) are parameters of the basis function @c. What is important is the fact that only for p = 1 are the cartesian coordinates eliminated from the matrix elements. For p = 1 the energy is invariant to the origin of the coordinate system and already known matrix elements are rederived [l I]. If p is limited to zero the well known matrix elements for GTo’s
appear. In general every basis function
can have its own parameter p_ In order to avoid very extensive variations we used only one parameter p for all ba-
sis functions. Two resuhs wiII be of particular interest: the variation of the energy with p and the energy coordinate origin dependence.
339
Voiurne57. number 3
CIfEMIcALP~SIcS
YXITERS
1 August 1978
3_ ResuIts The first system we have studied is H, molecule with the magnetic field perpendicular to the molecular axis. Double-zeta basis set 1121 implemented with poiarization functions was used_ Fig. 1 gives the v&es for the susceptiibiIit--. The explicit form of the function used to evaluate SusceptiiiIity when the coordinate origin is in the center of the mass is
where 2, and 2, are the coordinates of the hydrogen atoms with vaIues -0.7 au and i-O.7 au_ The basis set of the described form has been proposed by Epstein [ 13] and used on one-electron model sjrstem. Variation of the energy withy (fig_ 1, curve a) is very small, but interesting enough the energy with GTO’s is at lower value than with GIAO’s, CkarIy the statement that GIAO’s are a simple extension of GTO’s does not hold. The minima of the energies (corresponding to the lowest susceptibility vahxes) are at p = 0.927 til = -4.0736 X low6 cgs) and at p = 0396 (Q = 4_W78 X 10e6 cgs) for curves a and b (fig_ l), respectively. The susceptibility calculated with GIAO’s is x1 = 4.0771 X 1O-6 cgs. The HF moIecuIe has been treated with two basis sets. The results with the minimal basis set [14] F (2s2p/ 2s2p) and H (2s/ls), giving acceptabIe values for the susceptibility [8 ] , are depicted in fig. 2. As in the Hz molecule, aIso in HF molecule GTO’s give Iower energies than the GIAO ones (curves a, b). Motia [15] has argued tLt the second order (magnetic field as perturbation) SCF energy is higher than the second order Hartree-Fock one. He poWed out that the coordinate system should be determined by minimization of the second order SCF energy. in our case this means that the coordinate system where the lowest energy is obtained is the preferred one. The above described basis set has been used to calculate chemical shield&s (table 1). The conclusion from these results can be that GTo’s can be used for chemicaI shielding caiculations but the coordinate system should not be on the nucleus of the interest_ The coordinate system based or, the center of the mass (third coordinates in table 1) or on the almost identicaI electronic centroid has no preference to other coordinate sysrem centered on H or F atom_
Fg- l-Susceptibility of X-I=as a function of the parameterp. Coordinates (a) 21 = -0.7 all, 2, = 0.7 au; (5) 21 = 0 au, 22 = 1.4 au.
340
FE. 2. Sus~ptioilit~ of HF as a fumtion of the parameterp. coordinates (a) ZF = 0 au, ZH = l-733 au; (b) ZF = -0.1733 au, ZH = 1.5597 au; (c) .ZF = -1.733 au, ZH = 0 au.
vahune 57. number 3
CHEMKxLPHYsrCS
1 August 1978
LJZMERS
Table 1 Chemical shieIdings u @pm) Coordkates =F
co> (-1.733)
(-0.1733)
(au)
P
OF
=H
(0)
1 05 0 ;5
408-7 (40.58) a) 330.4 252.2 (365.8) 408.7 (405.8)
295 (300) aI 29.3 29.0 (23.9) 295 (30.0)
(1.5597)
0 0
-%I (1.733)
447.9
6f.5
487.0 (423.4) 275.6
93.5 (102.4) 35.4
a) Dunning basis set [16].
The HF molecule has been treated also by extended basis set of Dunning 1161 F (9s5p/3s2p) and H (4s/2s). This basis set gives, in the absence of a magnetic field, an energy value which is 0.05019 au off the l&-tree-Fock limit. One shonld then expect that the p dependence of the energy or the susceptibility will be only slight_ The results are contrary to these expectations. For p = 0 the susceptiiilities til) are -10.333 X 1W6 cgs and -26.899 X IO-6 cgs for the coordinate systems centered on F or H. In all calculations the coordinate system origins were shifted along molecular axis z. If the origin is shifted more widely in other directions too the results with GTO’s are compIetely -unacceptable_Shifting the coordinates of HF molecub for 10 an in they direction the GTO basis set gives susceptibility (x1) -257.34 X iW6 cgs to be compared with -10.509 X lo-6 cgs ifCiAO’s are used.
4. Conclusion The results show that the GIAO basis is by no means the best basis set for susceptibility calculations_ The restricted variation as performed in this work can give lower energy than with the GIAO basis set. The calculation scheme employed here can be also used, following the suggestion of Moccia ClSj, to determine *&e origin of the coordinate system. In general this is not in the center of the mass or on the electronic centroid. In an arbitrary coordinate system the use of the GIAO basis set is to be preferred, In a many-atomic molecular system such a situation is normal.
[l]
B.R. Applemanand BD. Dailey, in: Advances in magnetic resonance, Vol. 7, ed. J.S. Laugh (Academic Press, New York,
1974). 121 G.P. Arri@ni,
hf. Maestro and R. Moccia, J. Chem. Phys. 49 (1968) 882; G.P. Arr&+ini and C. Guidotti,them. Phys Letters 6 (1970) 435.
[3] A3. SadIej, Chem. Phys. Letters 36 (1975) 129. [Cl F. London, J. Phys. Radium 8 (1937) 397. [SI S-T. Epstein,3. them. Phys. 58 (1973) 1592_ 161 E. Da&par& Chem. Phys, Letters 47 (1977) 279. [7] R. Ditchfield,J. Chem,Phys. 56 (1972) 5688;Chem. 65 (1976) 3123;Chem. Phys. 13 (1976) 187.
Phys. Letters IS (1972) 203; Mol. Phys. 27 (1974) 789; J. Chem. Phys
341
Vohtme 57. nu~~bez 3
CEiE?ulCAL?liYSICS
LmTERS
[SJ M.. HIadnik. D. Fumpernik, M_ &xe-r and A- AZiuan, Chem. Phys. Letters 42 (X976) 361; 44 (1976) 2. h’aturfors&_ 31a (1976) $727; 32a (X977) 411; &f. hxcer ad A. AZinan, Phys- Rev- Al6 (1977) 475. [9 j BX Appclmul, T. Tokuhiro, G. FraenkeZ and C-W. Kern, J_ Chem. P&ys. 60
342
1 Augist 1978 58; 48 (1977)
139;
SOME CALCULATIONS
M_ ZAI~~ER*, Brernical Instihtre
WITH MAGNETIC
D_ PUMPERS Boris Kid&,
1 August 1978
PHYSICS LETrE.E.
CHEMICAL
FiELD
DEPENDENT
ORBITALS
and A. A&MN Ljubljana, Yugoshv~
Received 13 ApriI 1978
Calculatious with magnetic field dependent orbit&, which include usual cartes& gaus.Gans md London’s tieid dependent orbit& as speciaI cases, are presentzxi_Magnetic suscep?lbiiity and magnetic shielding of Hz and HF moIecuIer are calculated within the fiite perturbation Hartree-Fock LCAO method.
1. introduction A theoretical description of molecular magnetic susceptibilities and shielding constants has been a difficult problem mainly because of the gauge invariance of the vector potential of the external magnetic field [l] . Finite basis set calculations show a large dependence on the origin of the vector potential [2] and even for elaborate basis sets the computed second order molecular magnetic properties reveal a considerable gauge origin dependence [3]. As a way of avoiding this difficulty London [4] suggested the use of a‘particular form of field dependent orbit& for the basis set (nocalkd gauge invariant atomic orbitals - GIAO’s). Since then London orbitals have been widely used and the discussion about their physical meaning and applicability in calculations of magnetic properties is going on_ Though such a basis set forces the gauge invariance of the energy, a problem with current conservation appeared [5,6] _ Epstein also pointed out [S] that the constant vector potential in the phase factor could be effectively varied to preserve the translational invariance and to remove some non-physical properties of current. Despite this problem London orbitals are considered to have the advantage of being optimal and correct through first order in magnetic field density for the one-center problem- Using these orbitals good results f7,8] have been obtained for magnetic shieklings and susceptibilities. Some authors are however inclined to the idea that the use of magnetic field dependent orbit& leads to the appropriate extension of the virtual spectra [9]. A very recent application of the field dependent orbitals is by Sadlej [lo] who did calculations of second order electric properties ofmolecules with considerable ,zsccess. The purpose of this paper is to present calculations of magnetic susceptibility and shiel ‘ing in the frame of the field dependent orbitals, which include a variational paramerer p in the phase factor. For & ecial values of these factors they redze to London orbitals @ = 1) and to the gaussian basis set @ = 0). One result of this calculation is that in some cases London orbit& give a worse susceptiiility than the field independent gaussian type orbitals (GTWs)_
2. Theory The -kinetic part of the one-electron
-totian
* Presentaddress: Term&a. Ljtiljana, Y~gosIavia. 338
for the system
in an external
magnetic field is
CHEMICAL PKYSICS LErlERs
VoIume 57. num’oer3
HI = -(fi2f2nz)V2
- (iefi/rn)A.V
1 August 1978
f (e2/2m)A2_
(1)
The vector potential is taken to be A(r) = $B x (r - RG). where RG is defmed ro some arbitrary origin. The magnetic field dependent orbital centered at nuclear site C is @c = x&$$
exp (-o$)
exp C-(iep/WC
-r] -
!a
The matrix elements of H, are given zs a sum of three terms: <@&Zr@c> = -(ti2/2m)KDc
+ (iefi/2m)L,c
f (e2/h)QW,
(3)
where (omitting indices D and C) K = 402 [S(Zi2) + S(m+2) + S(n+2)]
- a[4(Z+m+n) + 61 S
+ Z(Z- 1) S(Z-2) f m(??z- 1) S(m-2) L =BJ?rrS(??z-l,n+l)
+m(l-p)ZccS(m-l)
-n(l-p)YccS(n-1)
+n(l-p)XCGS(“-l)
S(Z-i-1) - Xcc
+Z(l-p)YoS(Z-l)
--m(l-_p)XcGS(m-l)
-Z,,S(m+l)]] - Ls(Z-l,n+l)
S(Z- 1) + 24 l--p) 12,
+B,{ZS(Z-l,m+l)
(4)
-RzS(?n+l,n-1)
+ 2&&-p)[Y~&?2+1)
+By{nS(Z+l,n-l) - Z(l-p)Zcc
+ n(?z- 1) S(ll-2),
S(n+l)]
)
-mS(Z+l.m-1)
+20(l-P)[XcGS(m+l)
- Y,S(Z+-l)]],
Q=Z3;{S(m+2)+S(n+2)+2(1--p)[YccS(m+l)+&S(n+l)] +B;
{S(n+2)
+S(z+2) f 2(1-p)[&
+ B;
{S(Z+2) + S(m+2)
- 2BxBz
+(I-~)~(Y&+zzy;)S}
S(?zH) +xcc
+ (I-P)[Z&
S(Z+l) +Xcc
S(m+l)]
S(Z+l) + XcG S(n+l)]
+fl--p)[&$m+l)
In these expressions S is the overlap integral and X,,
+Qs)
S(Z+1)] + ( 1-p)2(z&
+ 2( 1 -p) [XCc S(Z+l) i- Ycc S(m+l)]
- 2B,B,, @(I+1 ,m+l) + ( l-&[YcG
(5)
f ( l-~)~
(X&
+ (‘-P)~X~&~ + ( 1-p)2X&cc
+X&(n+l)]
(or YcG, Zcc)
+(:-p)2YCCZCG
+ Y&)
S3
S] S] S)
.
(6)
is defmed as
All parameters (o, Z,m, rz, R,, p) that appear in eqs. (4)-(6) are parameters of the basis function @c. What is important is the fact that only for p = 1 are the cartesian coordinates eliminated from the matrix elements. For p = 1 the energy is invariant to the origin of the coordinate system and already known matrix elements are rederived [l I]. If p is limited to zero the well known matrix elements for GTo’s
appear. In general every basis function
can have its own parameter p_ In order to avoid very extensive variations we used only one parameter p for all ba-
sis functions. Two resuhs wiII be of particular interest: the variation of the energy with p and the energy coordinate origin dependence.
339
Voiurne57. number 3
CIfEMIcALP~SIcS
YXITERS
1 August 1978
3_ ResuIts The first system we have studied is H, molecule with the magnetic field perpendicular to the molecular axis. Double-zeta basis set 1121 implemented with poiarization functions was used_ Fig. 1 gives the v&es for the susceptiibiIit--. The explicit form of the function used to evaluate SusceptiiiIity when the coordinate origin is in the center of the mass is
where 2, and 2, are the coordinates of the hydrogen atoms with vaIues -0.7 au and i-O.7 au_ The basis set of the described form has been proposed by Epstein [ 13] and used on one-electron model sjrstem. Variation of the energy withy (fig_ 1, curve a) is very small, but interesting enough the energy with GTO’s is at lower value than with GIAO’s, CkarIy the statement that GIAO’s are a simple extension of GTO’s does not hold. The minima of the energies (corresponding to the lowest susceptibility vahxes) are at p = 0.927 til = -4.0736 X low6 cgs) and at p = 0396 (Q = 4_W78 X 10e6 cgs) for curves a and b (fig_ l), respectively. The susceptibility calculated with GIAO’s is x1 = 4.0771 X 1O-6 cgs. The HF moIecuIe has been treated with two basis sets. The results with the minimal basis set [14] F (2s2p/ 2s2p) and H (2s/ls), giving acceptabIe values for the susceptibility [8 ] , are depicted in fig. 2. As in the Hz molecule, aIso in HF molecule GTO’s give Iower energies than the GIAO ones (curves a, b). Motia [15] has argued tLt the second order (magnetic field as perturbation) SCF energy is higher than the second order Hartree-Fock one. He poWed out that the coordinate system should be determined by minimization of the second order SCF energy. in our case this means that the coordinate system where the lowest energy is obtained is the preferred one. The above described basis set has been used to calculate chemical shield&s (table 1). The conclusion from these results can be that GTo’s can be used for chemicaI shielding caiculations but the coordinate system should not be on the nucleus of the interest_ The coordinate system based or, the center of the mass (third coordinates in table 1) or on the almost identicaI electronic centroid has no preference to other coordinate sysrem centered on H or F atom_
Fg- l-Susceptibility of X-I=as a function of the parameterp. Coordinates (a) 21 = -0.7 all, 2, = 0.7 au; (5) 21 = 0 au, 22 = 1.4 au.
340
FE. 2. Sus~ptioilit~ of HF as a fumtion of the parameterp. coordinates (a) ZF = 0 au, ZH = l-733 au; (b) ZF = -0.1733 au, ZH = 1.5597 au; (c) .ZF = -1.733 au, ZH = 0 au.
vahune 57. number 3
CHEMKxLPHYsrCS
1 August 1978
LJZMERS
Table 1 Chemical shieIdings u @pm) Coordkates =F
co> (-1.733)
(-0.1733)
(au)
P
OF
=H
(0)
1 05 0 ;5
408-7 (40.58) a) 330.4 252.2 (365.8) 408.7 (405.8)
295 (300) aI 29.3 29.0 (23.9) 295 (30.0)
(1.5597)
0 0
-%I (1.733)
447.9
6f.5
487.0 (423.4) 275.6
93.5 (102.4) 35.4
a) Dunning basis set [16].
The HF molecule has been treated also by extended basis set of Dunning 1161 F (9s5p/3s2p) and H (4s/2s). This basis set gives, in the absence of a magnetic field, an energy value which is 0.05019 au off the l&-tree-Fock limit. One shonld then expect that the p dependence of the energy or the susceptibility will be only slight_ The results are contrary to these expectations. For p = 0 the susceptiiilities til) are -10.333 X 1W6 cgs and -26.899 X IO-6 cgs for the coordinate systems centered on F or H. In all calculations the coordinate system origins were shifted along molecular axis z. If the origin is shifted more widely in other directions too the results with GTO’s are compIetely -unacceptable_Shifting the coordinates of HF molecub for 10 an in they direction the GTO basis set gives susceptibility (x1) -257.34 X iW6 cgs to be compared with -10.509 X lo-6 cgs ifCiAO’s are used.
4. Conclusion The results show that the GIAO basis is by no means the best basis set for susceptibility calculations_ The restricted variation as performed in this work can give lower energy than with the GIAO basis set. The calculation scheme employed here can be also used, following the suggestion of Moccia ClSj, to determine *&e origin of the coordinate system. In general this is not in the center of the mass or on the electronic centroid. In an arbitrary coordinate system the use of the GIAO basis set is to be preferred, In a many-atomic molecular system such a situation is normal.
[l]
B.R. Applemanand BD. Dailey, in: Advances in magnetic resonance, Vol. 7, ed. J.S. Laugh (Academic Press, New York,
1974). 121 G.P. Arri@ni,
hf. Maestro and R. Moccia, J. Chem. Phys. 49 (1968) 882; G.P. Arr&+ini and C. Guidotti,them. Phys Letters 6 (1970) 435.
[3] A3. SadIej, Chem. Phys. Letters 36 (1975) 129. [Cl F. London, J. Phys. Radium 8 (1937) 397. [SI S-T. Epstein,3. them. Phys. 58 (1973) 1592_ 161 E. Da&par& Chem. Phys, Letters 47 (1977) 279. [7] R. Ditchfield,J. Chem,Phys. 56 (1972) 5688;Chem. 65 (1976) 3123;Chem. Phys. 13 (1976) 187.
Phys. Letters IS (1972) 203; Mol. Phys. 27 (1974) 789; J. Chem. Phys
341
Vohtme 57. nu~~bez 3
CEiE?ulCAL?liYSICS
LmTERS
[SJ M.. HIadnik. D. Fumpernik, M_ &xe-r and A- AZiuan, Chem. Phys. Letters 42 (X976) 361; 44 (1976) 2. h’aturfors&_ 31a (1976) $727; 32a (X977) 411; &f. hxcer ad A. AZinan, Phys- Rev- Al6 (1977) 475. [9 j BX Appclmul, T. Tokuhiro, G. FraenkeZ and C-W. Kern, J_ Chem. P&ys. 60
342
1 Augist 1978 58; 48 (1977)
139;