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Longitudinal vector potentials for molecular magnetic properties
THEO CHEM Journal of Molecular Structure (Theochem) 336 (1995) l-5
Longitudinal
vector potentials for molecular magnetic properties Pa010 Lazzeretti
Dipartimento
di Chimica dell’iYniversit6
degli Studi di Modena,
Via Campi 183. 41100 Modena, Italy
Received 8 November 1994; accepted 22 November 1994
Abstract A gauge transformation of the transverse Coulomb vector potential is studied to define a new vector potential with a non-vanishing component in the direction of magnetic field. In this longitudinal gauge the diamagnetic contribution to magnetic susceptibility is an isotropic tensor, whereas the diagonal and average diamagnetic contributions to nuclear magnetic shielding are the same as in the Coulomb gauge. The paramagnetic contributions to magnetic properties undergo gauge transformation, in such a way that total niagnetic tensors are left invariant, provided that a series of sum rules involving the virial tensor operator is satisfied.
1. Introduction Within current quantum mechanical theories [1,2] molecular magnetic properties are written as sums of diamagnetic and paramagnetic contributions. This partition is, to some extent, arbitrary and cannot be attached a real physical meaning. Usually, the molecular interaction Lagrangian and the relative Hamiltonian are written in terms of the Coulomb vector potential: in fact, in a change of gauge, diamagnetic and paramagnetic terms interconvert, and only their sum is left invariant. Gauge transformations corresponding to a change of coordinate origin have been the most investigated [3]. A few more interesting changes of gauge have been put forward. For instance, the paramagnetic contributions can be evaluated within a “torque formalism” [4], bearing some relationships to the Henneberger method [5,6]. Within the Landau prescription [7] for the vector potential, irrespective of the coordinate system [3,8], the dia-
magnetic contribution to susceptibility is a diagonal second-rank tensor, and the diamagnetic contribution to nuclear magnetic shielding is represented by a second-rank asymmetric tensor with only six non-vanishing components, instead of nine as in the Coulomb gauge (in the absence of molecular symmetry). The present paper is aimed at studying the properties of a gauge transformation by which the transverse Coulomb vector potential is replaced by a new vector potential with a longitudinal component, i.e. non-vanishing along the direction of the magnetic field. As a result of this change of gauge, the diamagnetic contribution to susceptibility becomes isotropic, albeit non-diagonal, for any molecule of arbitrary symmetry, i.e. characterized by three equal diagonal components as in atoms. However, the diagonal components of the diamagnetic contribution to nuclear magnetic shielding are the same as in the Coulomb gauge. The paramagnetic contributions undergo corresponding transformations, in
0166-1280/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0166-1280(94)04106-7
2
P. LazzerettijJournal
ofMolecularStructure
such a way that total properties are the same as in the Coulomb gauge. A series of sum rules is worked out as constraints of gauge invariance. They turn out to be very general quantum mechanical relationships, e.g. hypervirial theorems for the virial tensor operator [9]. As such, they are fulfilled by exact eigenfunctions to any model Hamiltonian; in the case of approximate wavefunctions, a numerical check of these constraints provides valuable criteria for assessing the quality of a calculation.
(Theochem)
336 (1995) I-5
and the electronic states of the molecule are its eigenfunctions ]j) (in particular, the reference state is indicated by 1~)). In the presence of a magnetic field, the total Hamiltonian is represented by a truncated Taylor series Hy=Ho+H$+&H$B
(5)
where an interaction Hamiltonian, containing terms of first and second order in the field, has been introduced, i.e.
2. Gauge transformation The divergenceless Coulomb vector potential is a transverse field, defined via the equation A’=iBxr=A,
V-AI=0
and
(7) The total angular momentum operator is L = Cy=,I, and the virial tensor operator is defined (cf. Ref [9]) as
(2)
and will examine the definitions of magnetic properties in the new gauge Ay = A’ _t VA’
(6)
2m,c
(1)
where B = V x A’, the induction of the external time independent and spatially uniform magnetic field is invariant in a gauge transformation A’ --t A’ + VA, with A = A(r), an arbitrary scalar function of coordinates. We will study in particular a simple gauge with a longitudinal component AlI, i.e. such that V x Ali = 0, introducing the gauge function AT = $(&x2 + ByyZ + &z2)
=eB-L
and its matrix elements can be easily evaluated using the equations [lo]
(3)
To this end, let us now consider a molecule with n electrons and N nuclei. We denote by ri, pi and 1; = ri x pi, position, linear and angular momentum of the i electron, with charge e and mass m,, and by RI and Zle the position and charge of the Zth nucleus. The unperturbed Born-Oppenheimer Hamiltonian of the electrons is
(9) 2r,pfl = Q&
+ if&, + ?
[HO,(rarp)]
(10) (11)
3. Magnetic susceptibility
(4)
Adopting the notation of Ref. [3], diamagnetic and paramagnetic contributions to susceptibility
P. LazzerettilJournal
are derivatives energy:
of the second-order
of Molecular
Structure
molecular
(Theochem)
P"r
X xy
- ~~~“((al”~~l’)(‘lr,la)) -
e
pY_ Xa,0 ---
@wp
dB,dB,
(14
PLY XXX
3
336 (1995) l-5
Ja
J#a
- ~~~“((alL’li)(il”rxla)i e J#Q la _
(13)
wp= -~~~,~‘(UlH’l’lj)(jiH(l)lU) (14) Jfa
In the reference state 1~)the diamagnetic contributions to the susceptibility in the transformed gauge become dY xxx
=
dbe d2 =x2; xyy
.e2 =-3m,c2
dY-__-XXY
1 d4P =JXaa
For molecular eigenfunctions satisfying the hypervirial theorem [9], it is found via Eqs. (9) and (11) that
n
(Ic rfI) a
(15)
a
i=l
-~(a~$Cz~-~z-vJi~a)
CL 4m,c2
(16)
4m,c
(other tensor components are obtained by cyclic permutation of the indices x, y and z). They are related to analogous quantities in the Coulomb gauge [31
x:7
=$(a1
(I I)
aex:a
(22)
i=l
hx..$.d]
ia)
=O
(23)
(17) by the equations dY xxx
=
dV xxx
The paramagnetic contributions PL^y
X08 p4p= x:; + x$” + x,0 where
are
Eqs. (22)-(25) are not, in general, fulfilled by approximate wavefunctions although (23) and (24) may be satisfied for other reasons, for instance, by symmetry. Accordingly the degree to which they are obeyed in actual calculations will give indications on the overall accuracy. Therefore, only in the limit situation of wavefunctions obeying the hypervirial theorem (1 l), will total susceptibility in the new gauge (3) be the same as in the Coulomb gauge, i.e.
(26)
P. LazzerettilJournal of Molecular Structure (Theochem)
4
336 (1995) 1-5
4. Nuclear magnetic shielding
In the presence of a nuclear magnetic dipole pI, the interaction Hamiltonian contains the extra terms _ pZaB”
-_
&I’)
From the cross term of Hamiltonians H(l) and see Eq. (14) for the paramagnetic interaction energy, the shielding tensor is written H(“),
(36)
(27)
where the Coulomb term is
where
A: = : ~a~~~&
(28)
is the vector potential acting on electron i due to such a nuclear dipole, and Eiy
=e
and the other quantities are
cy - RI, Iri - W3
is the electric field of electron i on nucleus I. The operator for the magnetic field of electrons on nucleus Z in the absence of external magnetic field is
Using a definition analogous to Eq. (13) for the cross term of diamagnetic interaction energy, diamagnetic contributions to the shielding of nucleus Z become
where
=
(32) is the corresponding gauge, and
quantity
in the Coulomb
(34)
-L(a~$&zla) 2m,c2
(38) The expectation values in the right-hand sides of Eqs. (38) are obtained via identity (11) provided that the corresponding hypervirial theorem for the virial tensor o erator is satisfied. Therefore, only in P this case A:,” = -A$?, insuring gauge invariance of nuclear magnetic shielding under the gauge transformation induced by function (2).
P. LazzerettijJournal of Molecular Structure (Theochem)
5. Constraints for origin independence Theoretical total magnetic properties must be independent of the origin of the coordinate system. Therefore variations of paramagnetic and diamagnetic contributions should exactly cancel under a gauge transformation corresponding to an arbitrary shift d of origin r’ -+ r” =r’+d
336 (1995) l-5
5
i.e. the same as those found in a translation of the Coulomb gauge [3]. The nuclear magnetic shieldings in the new gauge are obtained by Eqs. (31) (36) and AdI-ff
-YY
= -
&
(ul E;lu) = -A;iy
(49)
e
(39
Conditions for translational invariance in the gauge (3) bring in additional sum rules, compared to the cases of Coulomb [3] and Landau vector potential [8]. In any event the same constraints can be obtained allowing for the transformation of Coulomb gauge induced by the function A2 = $ (B,x& + BYydY + B,zdi)
(51)
The sum rules for invariance are again the same as in the Coulomb gauge [3],
(40)
In the new gauge the magnetic susceptibilities are
dY XXY
e2
dV
dY
XXX
=
xx.y - 44-c’
=
d’8 XXY
(41)
-nd:
Acknowledgements (42)
(43)
Financial support to this work from the Italian Ministry of Public Education (MPI) and Consiglio Nazionale delle Ricerche (CNR) is gratefully acknowledged.
(44)
References (45)
-
4m2c2 e
KL
pY)-,dy
+
CL,> J’x)L,dxl
(46)
The invariance sum rules are
(47)
= m,niT,p
(48)
[1] J.H. Van Vleck, The Theory of the Electric and Magnetic Susceptibilities, Oxford University Press, 1932. 121 N.F. Ramsey, Phys. Rev., 78 (1950) 699; 86 (1952) 243; Molecular Beams, Oxford University Press, London, 1956. [3] P. Lazzeretti, Adv. Chem. Phys., 75 (1987) 507. [4] P. Lazzeretti and R. Zanasi, Phys. Rev. A, 32 (1985) 2607. [5] W.C. Henneberger, Phys. Rev. Lett., 21 (1968) 838. [6] C. Cohen-Tannoudji, J. DuPont-Rot and G. Grynberg, Photons and Atoms, J. Wiley, New York, 1989. [7] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, 4th revised English edn., Pergamon, Oxford, 1979. [8] 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. [9] S.T. Epstein, The Variation Method in Quantum Chemistry, Academic Press, New York, 1974. [IO] P. Lazzeretti, Theor. Chim. Acta, 87 (1993) 59.
Longitudinal
vector potentials for molecular magnetic properties Pa010 Lazzeretti
Dipartimento
di Chimica dell’iYniversit6
degli Studi di Modena,
Via Campi 183. 41100 Modena, Italy
Received 8 November 1994; accepted 22 November 1994
Abstract A gauge transformation of the transverse Coulomb vector potential is studied to define a new vector potential with a non-vanishing component in the direction of magnetic field. In this longitudinal gauge the diamagnetic contribution to magnetic susceptibility is an isotropic tensor, whereas the diagonal and average diamagnetic contributions to nuclear magnetic shielding are the same as in the Coulomb gauge. The paramagnetic contributions to magnetic properties undergo gauge transformation, in such a way that total niagnetic tensors are left invariant, provided that a series of sum rules involving the virial tensor operator is satisfied.
1. Introduction Within current quantum mechanical theories [1,2] molecular magnetic properties are written as sums of diamagnetic and paramagnetic contributions. This partition is, to some extent, arbitrary and cannot be attached a real physical meaning. Usually, the molecular interaction Lagrangian and the relative Hamiltonian are written in terms of the Coulomb vector potential: in fact, in a change of gauge, diamagnetic and paramagnetic terms interconvert, and only their sum is left invariant. Gauge transformations corresponding to a change of coordinate origin have been the most investigated [3]. A few more interesting changes of gauge have been put forward. For instance, the paramagnetic contributions can be evaluated within a “torque formalism” [4], bearing some relationships to the Henneberger method [5,6]. Within the Landau prescription [7] for the vector potential, irrespective of the coordinate system [3,8], the dia-
magnetic contribution to susceptibility is a diagonal second-rank tensor, and the diamagnetic contribution to nuclear magnetic shielding is represented by a second-rank asymmetric tensor with only six non-vanishing components, instead of nine as in the Coulomb gauge (in the absence of molecular symmetry). The present paper is aimed at studying the properties of a gauge transformation by which the transverse Coulomb vector potential is replaced by a new vector potential with a longitudinal component, i.e. non-vanishing along the direction of the magnetic field. As a result of this change of gauge, the diamagnetic contribution to susceptibility becomes isotropic, albeit non-diagonal, for any molecule of arbitrary symmetry, i.e. characterized by three equal diagonal components as in atoms. However, the diagonal components of the diamagnetic contribution to nuclear magnetic shielding are the same as in the Coulomb gauge. The paramagnetic contributions undergo corresponding transformations, in
0166-1280/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0166-1280(94)04106-7
2
P. LazzerettijJournal
ofMolecularStructure
such a way that total properties are the same as in the Coulomb gauge. A series of sum rules is worked out as constraints of gauge invariance. They turn out to be very general quantum mechanical relationships, e.g. hypervirial theorems for the virial tensor operator [9]. As such, they are fulfilled by exact eigenfunctions to any model Hamiltonian; in the case of approximate wavefunctions, a numerical check of these constraints provides valuable criteria for assessing the quality of a calculation.
(Theochem)
336 (1995) I-5
and the electronic states of the molecule are its eigenfunctions ]j) (in particular, the reference state is indicated by 1~)). In the presence of a magnetic field, the total Hamiltonian is represented by a truncated Taylor series Hy=Ho+H$+&H$B
(5)
where an interaction Hamiltonian, containing terms of first and second order in the field, has been introduced, i.e.
2. Gauge transformation The divergenceless Coulomb vector potential is a transverse field, defined via the equation A’=iBxr=A,
V-AI=0
and
(7) The total angular momentum operator is L = Cy=,I, and the virial tensor operator is defined (cf. Ref [9]) as
(2)
and will examine the definitions of magnetic properties in the new gauge Ay = A’ _t VA’
(6)
2m,c
(1)
where B = V x A’, the induction of the external time independent and spatially uniform magnetic field is invariant in a gauge transformation A’ --t A’ + VA, with A = A(r), an arbitrary scalar function of coordinates. We will study in particular a simple gauge with a longitudinal component AlI, i.e. such that V x Ali = 0, introducing the gauge function AT = $(&x2 + ByyZ + &z2)
=eB-L
and its matrix elements can be easily evaluated using the equations [lo]
(3)
To this end, let us now consider a molecule with n electrons and N nuclei. We denote by ri, pi and 1; = ri x pi, position, linear and angular momentum of the i electron, with charge e and mass m,, and by RI and Zle the position and charge of the Zth nucleus. The unperturbed Born-Oppenheimer Hamiltonian of the electrons is
(9) 2r,pfl = Q&
+ if&, + ?
[HO,(rarp)]
(10) (11)
3. Magnetic susceptibility
(4)
Adopting the notation of Ref. [3], diamagnetic and paramagnetic contributions to susceptibility
P. LazzerettilJournal
are derivatives energy:
of the second-order
of Molecular
Structure
molecular
(Theochem)
P"r
X xy
- ~~~“((al”~~l’)(‘lr,la)) -
e
pY_ Xa,0 ---
@wp
dB,dB,
(14
PLY XXX
3
336 (1995) l-5
Ja
J#a
- ~~~“((alL’li)(il”rxla)i e J#Q la _
(13)
wp= -~~~,~‘(UlH’l’lj)(jiH(l)lU) (14) Jfa
In the reference state 1~)the diamagnetic contributions to the susceptibility in the transformed gauge become dY xxx
=
dbe d2 =x2; xyy
.e2 =-3m,c2
dY-__-XXY
1 d4P =JXaa
For molecular eigenfunctions satisfying the hypervirial theorem [9], it is found via Eqs. (9) and (11) that
n
(Ic rfI) a
(15)
a
i=l
-~(a~$Cz~-~z-vJi~a)
CL 4m,c2
(16)
4m,c
(other tensor components are obtained by cyclic permutation of the indices x, y and z). They are related to analogous quantities in the Coulomb gauge [31
x:7
=$(a1
(I I)
aex:a
(22)
i=l
hx..$.d]
ia)
=O
(23)
(17) by the equations dY xxx
=
dV xxx
The paramagnetic contributions PL^y
X08 p4p= x:; + x$” + x,0 where
are
Eqs. (22)-(25) are not, in general, fulfilled by approximate wavefunctions although (23) and (24) may be satisfied for other reasons, for instance, by symmetry. Accordingly the degree to which they are obeyed in actual calculations will give indications on the overall accuracy. Therefore, only in the limit situation of wavefunctions obeying the hypervirial theorem (1 l), will total susceptibility in the new gauge (3) be the same as in the Coulomb gauge, i.e.
(26)
P. LazzerettilJournal of Molecular Structure (Theochem)
4
336 (1995) 1-5
4. Nuclear magnetic shielding
In the presence of a nuclear magnetic dipole pI, the interaction Hamiltonian contains the extra terms _ pZaB”
-_
&I’)
From the cross term of Hamiltonians H(l) and see Eq. (14) for the paramagnetic interaction energy, the shielding tensor is written H(“),
(36)
(27)
where the Coulomb term is
where
A: = : ~a~~~&
(28)
is the vector potential acting on electron i due to such a nuclear dipole, and Eiy
=e
and the other quantities are
cy - RI, Iri - W3
is the electric field of electron i on nucleus I. The operator for the magnetic field of electrons on nucleus Z in the absence of external magnetic field is
Using a definition analogous to Eq. (13) for the cross term of diamagnetic interaction energy, diamagnetic contributions to the shielding of nucleus Z become
where
=
(32) is the corresponding gauge, and
quantity
in the Coulomb
(34)
-L(a~$&zla) 2m,c2
(38) The expectation values in the right-hand sides of Eqs. (38) are obtained via identity (11) provided that the corresponding hypervirial theorem for the virial tensor o erator is satisfied. Therefore, only in P this case A:,” = -A$?, insuring gauge invariance of nuclear magnetic shielding under the gauge transformation induced by function (2).
P. LazzerettijJournal of Molecular Structure (Theochem)
5. Constraints for origin independence Theoretical total magnetic properties must be independent of the origin of the coordinate system. Therefore variations of paramagnetic and diamagnetic contributions should exactly cancel under a gauge transformation corresponding to an arbitrary shift d of origin r’ -+ r” =r’+d
336 (1995) l-5
5
i.e. the same as those found in a translation of the Coulomb gauge [3]. The nuclear magnetic shieldings in the new gauge are obtained by Eqs. (31) (36) and AdI-ff
-YY
= -
&
(ul E;lu) = -A;iy
(49)
e
(39
Conditions for translational invariance in the gauge (3) bring in additional sum rules, compared to the cases of Coulomb [3] and Landau vector potential [8]. In any event the same constraints can be obtained allowing for the transformation of Coulomb gauge induced by the function A2 = $ (B,x& + BYydY + B,zdi)
(51)
The sum rules for invariance are again the same as in the Coulomb gauge [3],
(40)
In the new gauge the magnetic susceptibilities are
dY XXY
e2
dV
dY
XXX
=
xx.y - 44-c’
=
d’8 XXY
(41)
-nd:
Acknowledgements (42)
(43)
Financial support to this work from the Italian Ministry of Public Education (MPI) and Consiglio Nazionale delle Ricerche (CNR) is gratefully acknowledged.
(44)
References (45)
-
4m2c2 e
KL
pY)-,dy
+
CL,> J’x)L,dxl
(46)
The invariance sum rules are
(47)
= m,niT,p
(48)
[1] J.H. Van Vleck, The Theory of the Electric and Magnetic Susceptibilities, Oxford University Press, 1932. 121 N.F. Ramsey, Phys. Rev., 78 (1950) 699; 86 (1952) 243; Molecular Beams, Oxford University Press, London, 1956. [3] P. Lazzeretti, Adv. Chem. Phys., 75 (1987) 507. [4] P. Lazzeretti and R. Zanasi, Phys. Rev. A, 32 (1985) 2607. [5] W.C. Henneberger, Phys. Rev. Lett., 21 (1968) 838. [6] C. Cohen-Tannoudji, J. DuPont-Rot and G. Grynberg, Photons and Atoms, J. Wiley, New York, 1989. [7] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, 4th revised English edn., Pergamon, Oxford, 1979. [8] 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. [9] S.T. Epstein, The Variation Method in Quantum Chemistry, Academic Press, New York, 1974. [IO] P. Lazzeretti, Theor. Chim. Acta, 87 (1993) 59.