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A theory of magnetic vibrational circular dichroism
Volume
114, number
3
CHEMICAL
PHYSICS
A THEORY OF MAGNETIC VIBRATIONAL
LETTERS
CIRCULAR
1 March
1985
DICHROISM
Thomas H. WALNUT Department of Chemistry, Syracuse University, Syracuse, NY 13210, USA Received
28 August
1984;in
final form 6 December
1984
A theory of type B magnetic vibrational circular dichroism (MVCD) is developed and applied. The motion of the nuclei is treated classically and the momentum correlation between their motion and the electronic motion is treated explicitly.
Recently Keiderling and Devine [l-3] have measured the magnetic vibrational circular dichroism (MVCD) of a moderate number of simple molecules. In the present paper a theory of this phenomenon is developed which differs from previous theory [4-61 in that it explicitly considers momentum correlation between nuclear and electronic motion. Only type B MVCD (the part that arises from the change in the wavefunction) is the subject of this paper. The Hamiltonian operator for a molecule in a magnetic field is H= 7
(1/2m)
+ c (1/2M,) n
rpi
+
cc/C)
Ail2
[P, - (e/c>z,A,l*
+ v.
(1)
The symbols have their usual meaning. The first term is the electronic kinetic energy in a magnetic field, the second, the nuclear kinetic energy, and the third is the potential energy. Part of the second term can be used to form the following expression
and r refers to the electronic coordinates and R to the nuclear coordinates. The first term on the left is the standard vibrational kinetic energy term. Together with the terms in the electronic kinetic energy involvingp/ and with V, it forms H(0)tiBo, the unperturbed part of the problem. The third term on the right of eq. (2) yields effective local reduced electronic masses [7], and will not be considered further. The second term is the nuclear-electronic momentum correlation term and is the basis of the rest of the treatment. The basis of the method is to treat the electronic part of the problem as if each nucleus has simultaneously a discreet position and velocity. A convenient device for doing this is to find the electronic wavefunction, $,, that minimizes EL, in the expression
where duel refers to integration over only the electronic coordinates. It should be noted that in the BO approximation the minimizing $, is independent of @N.
The function @N that most simply introduces (constant) nuclear velocities v, is %N(v,R)=
where GBO, the Born-Oppenheimer GBO = Ge(y, R, @N(R)
wavefunction
is
exp i CM,V;R,/A (
II
)
,
the
(5)
where v represents the set of velocities, v, , and R, is the position of the rlth nucleus. These functions can be used to obtain $,, the velocity corrected exten-
(3)
0 009-2614/8.5/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
265
Volume
114, number
3
sion of the BO wavefunction.
CHEMICAL
PHYSICS
as a Fourier transform,
1 March 1985
LETTERS
Hcl>“$, = c nC1Ai*pj$e ,
(12a)
i H@J)$e where \lle@. v) is the minimizing wavefunction, xN(v) is the vibrational wavefunction in velocity space and N is the number of degrees of freedom. The complete solution is a weighted superposition of the solutions for each set of values of v. The set of velocities chosen for this discussion are those appropriate to a particular normal coordinate Qr. When aN(v, R) IS used T, the second term in eq. (2), becomes
= vPGc .
(12b)
= 0.
(12c)
and NJ)&
I$t,l) vanishes because there is no electronic cross term in B and v. MVCD of type B depends on $(I ,l); the effect is first order in B and u. The standard method of obtaining $(I 31) is from the equation (0) (1,1) = - E, ) G
(&‘j
_~(l.O)~(O,l)
_ytoJ)~(l,o)
(13) The Cartesian nuclear momentum operators P,, can be expressed in terms of the momentum operators Pk, the conjugates of the normal coordinates Qk, as P, =
ck bknPk .
(8)
In this expression the functions $(O,l) and $(l,o) are needed. Finding them can be avoided by changing the local frames of reference. The procedure for doing this is well known [8,9] and is equivalent to a gage transformation. Formally, the Hamiltonian, HM r, for the moving frame is formed by H MF = epiSHeiS
yielding (9)
(14)
where (15)
s = eflcfi - vmu/A When U, conforms
to Ql
c Un-bkn = V 6,, n
(10)
and, hence, T = @,(v, R) v PGe ,
(11)
where v is the constant (mass-weighted) velocity for Ql and P = P,. The problem to be solved is then the problem of finding wavefunction for the electronic ground state when the nuclear positions are given by the vectors R, and the nuclear velocities by II,. The minimizing function, Ge(y, v) is the lowest energy eigenfunction of He where, to first order in v and B (the magnetic field strength),
Actually, S represents two different superimposed moving frames, one involving the functionf’that simplifies the description of the magnetic field and the other involving the function u that describes the following of nuclear motion. Both yield first-order effects, and, consequently, their interaction is a second-order effect. The results of the transformation are ~~~’
+ Pj’(elc)
I!@$)
[(e/C)
(Ai + v,fJl
= VP - v c i
H’M’I;!’ = C
(Aj
+Vif)‘pi
)
(16a)
[2(-iF$u)*pi
[(e/c) (Aj + Vjn]
- hV;u]
,
(16b)
‘<-VviU)
j
t
266
(l/?im) i
He=H(,O)+~“O)+H(O~l)+H(l,l). Here H(O) is the electronic part of the BO Hamiltonian unperttrbed by the magnetic field,
=c
(ie/c?zj v[P,f]
(16~)
The formal procedure for snaking the optiInun1 choice of moving frame is to adjust the functionsJ’and u so that $@) and $$)cl) vanish (or at least are minimized). In physical terms, the nuclei are stationary with respect to the moving frames, and no further change in the wavefunction is required. The functions $ge”’ and $Gt;“) would vanish iffand u were chosen so that H(~c’)$~ and @$pl)$e vanished. At this point it will be assumed that this can be done. Consequently, the second-order equation, eq. (13), is transformed into an equation that is first order in uB, i.e. (@)
_&O’)
&l’
= -#&‘$/e
.
(17)
Solution of eq. (17) is greatly helped by further visualization of the meaning of eq. (16~). In the first term the quantity (e/c) (Ai + VJ) is the momentum, mum 2induced for one electron by the magnetic field, and -uViu is the moving frame velocity U, induced in the electronic motion by the nuclear motion. This is a valid point of view because the first-order equations obtained from eqs. (16a) and (16b) can readily be transformed to continuity equations involving U, and u, [7,10]. The contribution to H(,‘$‘) for one electron is then I$’Mi; l) = mu m’%
.
1 March 1985
CHEMICAL PHYSICS LETTERS
Volume 114, number 3
(18)
The interaction is the cross term in the combined kinetic energy for the two velocity fields. For a ground state, a first-order equation implies that the electron density will tend to increase where &de’) is negative and decrease where it is positive. It is positive where II,,, and u, have the same direction and negative where they are opposed. The velocity, u, depends on the phase and amplitude of the vibration. It is largest when the nuclei are moving at maximum speed. At these times the nuclei are at their equilibrium positions. The charge distortion that arises from H’del ) is then 90” out of phase with the usual charge distortion that arises during a vibration. The theory can now be applied to the simple case, the elu ring distortion mode of benzene. The situation at maximum distortion is symbolized by the ring at the right in fig. la. The ring on the left represents the situation l/4 cycle earlier. At this stage the nuclei are in their equilibrium positions and the usual dipole
t t H
>HYN\ H, Fig. 1. Diagrams describing MVCD (a) for the qu ring distortion mode of benzene and (b) for the e stretching mode of ammonia. The solid arrows represent uU, the electronic velocity induced by nuclear motion. The dashed arrows represent II,, the electronic velocity induced by the magnetic field. The circled + and - signs can be considered to indicate, respectively, that U, and 9, are parallel or antiparallel. They also can be considered to indicate the sipn of the induced second-order charge. The diagrams on the left represent the vibrations at the beginnings of their cycles (maximum speed, minimum displacement). The diagrams on the right represent the same vibrations l/4 cycle later (minimum speed, maximum displacement). The + and - siens describe the change in charge distribution in poing from the equilibrium configuration to maximum displacement. The solid continuous lines symbolize the contours of constant probability density.
vanishes. In order to form the dipole on the right, there must be a flow of electrons toward the bottom of the ring. The velocity field for this flow isu,. The electronic motion induced by the magnetic field circulates around the ring and is given by U, On the left side of the first ring U, and U, are parallel and mum -u, is positive; on the right they are an tiparallel and mu, -II” is negative. There is then an excess of charge (deficiency of electrons) on the left and a deficiency of charge on the right. This situation results in a weak transverse dipole followed l/4 cycle later by a strong vertical dipole. A rotating dipole is associated with circular dichroism, counterclockwise with left or positive circular dichroism and clockwise with right or negative circular dichroism. Fig. 1a shows an excess of clockwise flow and predicts negative circular dichroism in agreement with experiment It is possible to make some rough calculations of 267
Volume 114, number 3
CHEMICAL PHYSICS LETTERS
the circular dichroism for this vibration. The key is the fact that by using only the equation of continuity and the vibrational frequency, the current, and hence U, can be completely determined from ap/aQ. The vector u, can easily be obtained from the Larmor precession frequency. One can assume a one-electron interaction for H#c’) and solve eq. (17). In this case the six n-electrons of the ring are assumed to be the source of all the effect. and they are treated as particles on a circular track of radius Y. In the appendix the specific circular dichroism, B,/D,, is shown to be Be/Do = 64~r~rn~c~i%~/9i?~ ,
(19)
where V is the frequency in cm- 1 and the units of Be/Do are l/cm-l . The value of r is chosen such that the magnitude of the vector potential for the circular track is the same as that on a track that follows the carbon-carbon bonds in the ring, i.e.
s
A-ds=BA,,
(20)
where A, is the area of the ring. For benzene this yields 6&I\ = 31i2B 12/4
w>
(23)
On using 1= 1.39 A and V = 1480 cm-l it is predicted that B,/D, = -1.5 X lop5 per cm-l. The observed value is - 1.2 X 10-j per cm-l . The closeness of agreement is fortuitous; satisfactory agreement would lie within a factor of 2. A second example is the e stretching mode of NH,. In this case the nuclei contribute to the dipole moment, and the experimentally measured sign of ap/aQ [ 1 l] must be used in the theory. The dipole at maximum distortion is indicated on the right of fig. lb. The situation at the beginning of a vibrational cycle is shown on the left of fig. lb. The electrons tend to follow the nuclei and the magnetically induced electronic motion tends to follow contour lines of electron density. The induced out-of-phase dipole is predicted to point to the right, leading to 268
.
The quantitative expression for MVCD is (Aleft and the effect tends to be large when - Ari&%” A,, is small. For NH,, A,, is small. For NH,, A,, happens to be relatively small yielding a moderately large dichroism. A large value of A,, for the observed modes of paradichlorobenzene [3] provides an explanation other than symmetry for the fact MVCD is not observed for this compound. The prediction of strong symmetry dependence found in previous theory [4-61 does not appear in the theory given in this paper. At this point we return to the second term in H(d$ ) (cf. eq. (16~)). It can be thought of as compensating for the first term. Consider. as an example, an argon atom translating perpendicular to a uniform magnetic field. If the analysis that was done in fig. la for the benzene ring is applied to this case (an atom, including its electrons. moving down) a transverse dipole moment would be predicted, contrary to fact. For this system the appropriate gage function that keeps the vector potential centered on the nucleus of the atom (at R) is I’= -A-R
giving r= 3I121/2.
left circular dichroism. again in agreement with experiment
(21)
and for the ring 27rrlA 1= Brrr2 ,
1 March 1985
.
(24)
The second term is then a translational correction term and in this case it is readily apparent that it cancels the first term. It should be noted that in the treatment of the benzene molecule the gage function vanished everywhere, and only the first term contributed. Another problem to be considered is the vanishing of @dcu’ that was assumed in the development of eq. (17). This problem has been extensively investigated [ 10,121. In most cases, it is not possible to adjust the gage function so that dde”)$e vanishes everywhere. Fortunately the currents that are best approximated by using gage functions are those in the outer parts of molecules, regions where the largest contributions to MVCD are made. In principle the same considerations should be valid for the vanishing of die1 ) $e on adjusting II. In practice, for most vibrations non-vanishing circulation, which is an essential part of the treatment of magnetic effects, is only a minor consideration. It should then be expected that choosing u so that
1 March 1985
CHEMICALPHYSICS LETTERS
Volume 114, number 3
The dipole moment operator associated with the absorption of left circularly polarized radiation is
d$el ) $, nearly vanished would be a good approximation. This approach is consistent with that of using the equation of continuity (cf. eq. (27)) as was done in the treatment of the etu mode of benzene.
I*x+iy = -er(cos
Appendix
The usual procedure in treating vibrational absorption is to average the dipole moment over the electronic coordinates giving
The oscillating electronic (angular) density for one of the pairs of degenerate e,ti vibrations is pi = (6/27i) [I t ir sin 4 sin(wt)]
px+iJ> =
(25)
The velocity u, is u, = ViPi
I
(26)
where j is the current obtained tion of continuity,
by integrating
aj/aG =-$+/at. The quantity
the equa-
(27) u, is found to be
(35)
(36)
s PPx+iy d7el )
where p is the total electronic probability density. The part of p which makes a significant contribution to the circular dichroism is p=p~o~~p~l~+p~2~+p~3~=pj+p~2~,
(37)
where
t (#’
+ $(11’1))2 t (I@’
t @1))2]
+ 0 ((uB)~) .
u, = --aor sin @Icos(wt) t 0 (0”) The velocity u, is obtained sion frequency wL, i.e.
f$ + i sin 4)
(28)
from the Larmor preces-
(38)
From eqs. (25) (31), (34a, b, c), and (38) pc2) = 2B’sin @
(39)
and urn = WLY= eBr/2mc The Hamiltonian
The unperturbed
(30)
wavefunctions
are
I$~“) = n-l /2 sin $J,
cos f$
(31)
eqs. (17), (IS), (28) (29), and (30) yields
a2$,(1~1)/@2
+nf$i(L,L)
= _B’sin @ ruin) ,
(32)
where B’ = -amweBr The second-order
cos(ot)/fi2c
(33)
(40)
&+iv = -3aer [sin(wt) + 2iB/9] . In terms of the normal coordinate jugate momentum P, _ px+iy = -3Cer(Q + 2iDP/9w) ,
(41) Q and its con-
(42)
where CQ = a sin(wt) ,
(43)
CQ = CP = aw cos(wt) ,
(44)
D = B’/a cos(wt) ,
(45)
solutions of eq. (32) are
$b’>‘) = (27r)lj2 B' sin 4, 1//y)
p(l) = 6a cos 4 sin(wt) , yielding
a’/aQ
$60) = (271)-l i2,
Combining
(29)
operator for a particle on a ring is
tico) = -(fi2/2mrz)
$JJ’ = .-I/2
.
= d4B’/6)
li/$‘sl) = 7F1’2(~‘/6)
(-cos sin 24
(34b)
and C is a constant. For px_iy the sign of the second term would be negative. The transition moment between two neighboring vibrational states is found to be
(34c)
I%+
Pa) 24 - 3) )
,P@u dr, = s%c,
[-3erC(1
+ 2D/9)Q] av dr,, (46) 269
Volume
114. number
CHEMICAL
3
when the expression (47) is employed.
Since Aleft
is proportional
I(u + 11Eli,+izlu)2 and Arieht LUIA,,
to
to I(u + 1 IP~,i,lu)l*
= 4 X 2019 = 8rnweBr4/9A2c
To obtain
B,/D,
in I/cx-~
by eB/47rnzc2 yielding
it is necessary
a (48)
to divide
eq. (19).
References [ 1 ] T.A. Keiderling. J. Chem. Phys. 75 (1981) 3639. [2] T.R. Devine and T.A. Keiderlinf. J. Chem. Phys. 79 (1983) 5796.
270
PHYSICS
LETTERS
1 March 1985
[3] T.R. Devinc and T.A. Keiderling. J. Phys. Chem. 88 (1984) 390. [4] S.L. Cunningham and J.R. Hardy, Solid State Commun. 6 (1968) 769. 15 1 S. Datta and I’.S. Richardson, Intern. J. Quantum Chcm. 11 (I 977) 525. [6] L. Laus. V. Pultz. C. Marcott. J. Ovcrend and A. Mosco\vit/, J. Chem. Phys. 78 (1983) 4096. [7] T.H. Walnut and L.A. Nafie. J. Chem. Phys. 67 (1977) 1491. [8] S.T. Flpstein, .I. Chem. Phys. 58 (1973) 1592. 191 J. Overend, in: Vibrational intensities in infrared and Raman spectroscopy. eds. W.H. Person and G. Zerbi (Elsevier. Amsterdam, 1982) pp. 190-~202. [ 101 R.A. llepstrom and W.N. Lipscomb, Rev. Mod. Phys. 40 (1968) 354. [ 111 S. Abbate, S.L. Wunder and G. Zerbi, .I. Phys. Chem. 88 (1984) 600. [12] T.H. Walnut. J. Chem. Phys. 65 (1976) 3647.
114, number
3
CHEMICAL
PHYSICS
A THEORY OF MAGNETIC VIBRATIONAL
LETTERS
CIRCULAR
1 March
1985
DICHROISM
Thomas H. WALNUT Department of Chemistry, Syracuse University, Syracuse, NY 13210, USA Received
28 August
1984;in
final form 6 December
1984
A theory of type B magnetic vibrational circular dichroism (MVCD) is developed and applied. The motion of the nuclei is treated classically and the momentum correlation between their motion and the electronic motion is treated explicitly.
Recently Keiderling and Devine [l-3] have measured the magnetic vibrational circular dichroism (MVCD) of a moderate number of simple molecules. In the present paper a theory of this phenomenon is developed which differs from previous theory [4-61 in that it explicitly considers momentum correlation between nuclear and electronic motion. Only type B MVCD (the part that arises from the change in the wavefunction) is the subject of this paper. The Hamiltonian operator for a molecule in a magnetic field is H= 7
(1/2m)
+ c (1/2M,) n
rpi
+
cc/C)
Ail2
[P, - (e/c>z,A,l*
+ v.
(1)
The symbols have their usual meaning. The first term is the electronic kinetic energy in a magnetic field, the second, the nuclear kinetic energy, and the third is the potential energy. Part of the second term can be used to form the following expression
and r refers to the electronic coordinates and R to the nuclear coordinates. The first term on the left is the standard vibrational kinetic energy term. Together with the terms in the electronic kinetic energy involvingp/ and with V, it forms H(0)tiBo, the unperturbed part of the problem. The third term on the right of eq. (2) yields effective local reduced electronic masses [7], and will not be considered further. The second term is the nuclear-electronic momentum correlation term and is the basis of the rest of the treatment. The basis of the method is to treat the electronic part of the problem as if each nucleus has simultaneously a discreet position and velocity. A convenient device for doing this is to find the electronic wavefunction, $,, that minimizes EL, in the expression
where duel refers to integration over only the electronic coordinates. It should be noted that in the BO approximation the minimizing $, is independent of @N.
The function @N that most simply introduces (constant) nuclear velocities v, is %N(v,R)=
where GBO, the Born-Oppenheimer GBO = Ge(y, R, @N(R)
wavefunction
is
exp i CM,V;R,/A (
II
)
,
the
(5)
where v represents the set of velocities, v, , and R, is the position of the rlth nucleus. These functions can be used to obtain $,, the velocity corrected exten-
(3)
0 009-2614/8.5/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
265
Volume
114, number
3
sion of the BO wavefunction.
CHEMICAL
PHYSICS
as a Fourier transform,
1 March 1985
LETTERS
Hcl>“$, = c nC1Ai*pj$e ,
(12a)
i H@J)$e where \lle@. v) is the minimizing wavefunction, xN(v) is the vibrational wavefunction in velocity space and N is the number of degrees of freedom. The complete solution is a weighted superposition of the solutions for each set of values of v. The set of velocities chosen for this discussion are those appropriate to a particular normal coordinate Qr. When aN(v, R) IS used T, the second term in eq. (2), becomes
= vPGc .
(12b)
= 0.
(12c)
and NJ)&
I$t,l) vanishes because there is no electronic cross term in B and v. MVCD of type B depends on $(I ,l); the effect is first order in B and u. The standard method of obtaining $(I 31) is from the equation (0) (1,1) = - E, ) G
(&‘j
_~(l.O)~(O,l)
_ytoJ)~(l,o)
(13) The Cartesian nuclear momentum operators P,, can be expressed in terms of the momentum operators Pk, the conjugates of the normal coordinates Qk, as P, =
ck bknPk .
(8)
In this expression the functions $(O,l) and $(l,o) are needed. Finding them can be avoided by changing the local frames of reference. The procedure for doing this is well known [8,9] and is equivalent to a gage transformation. Formally, the Hamiltonian, HM r, for the moving frame is formed by H MF = epiSHeiS
yielding (9)
(14)
where (15)
s = eflcfi - vmu/A When U, conforms
to Ql
c Un-bkn = V 6,, n
(10)
and, hence, T = @,(v, R) v PGe ,
(11)
where v is the constant (mass-weighted) velocity for Ql and P = P,. The problem to be solved is then the problem of finding wavefunction for the electronic ground state when the nuclear positions are given by the vectors R, and the nuclear velocities by II,. The minimizing function, Ge(y, v) is the lowest energy eigenfunction of He where, to first order in v and B (the magnetic field strength),
Actually, S represents two different superimposed moving frames, one involving the functionf’that simplifies the description of the magnetic field and the other involving the function u that describes the following of nuclear motion. Both yield first-order effects, and, consequently, their interaction is a second-order effect. The results of the transformation are ~~~’
+ Pj’(elc)
I!@$)
[(e/C)
(Ai + v,fJl
= VP - v c i
H’M’I;!’ = C
(Aj
+Vif)‘pi
)
(16a)
[2(-iF$u)*pi
[(e/c) (Aj + Vjn]
- hV;u]
,
(16b)
‘<-VviU)
j
t
266
(l/?im) i
He=H(,O)+~“O)+H(O~l)+H(l,l). Here H(O) is the electronic part of the BO Hamiltonian unperttrbed by the magnetic field,
=c
(ie/c?zj v[P,f]
(16~)
The formal procedure for snaking the optiInun1 choice of moving frame is to adjust the functionsJ’and u so that $@) and $$)cl) vanish (or at least are minimized). In physical terms, the nuclei are stationary with respect to the moving frames, and no further change in the wavefunction is required. The functions $ge”’ and $Gt;“) would vanish iffand u were chosen so that H(~c’)$~ and @$pl)$e vanished. At this point it will be assumed that this can be done. Consequently, the second-order equation, eq. (13), is transformed into an equation that is first order in uB, i.e. (@)
_&O’)
&l’
= -#&‘$/e
.
(17)
Solution of eq. (17) is greatly helped by further visualization of the meaning of eq. (16~). In the first term the quantity (e/c) (Ai + VJ) is the momentum, mum 2induced for one electron by the magnetic field, and -uViu is the moving frame velocity U, induced in the electronic motion by the nuclear motion. This is a valid point of view because the first-order equations obtained from eqs. (16a) and (16b) can readily be transformed to continuity equations involving U, and u, [7,10]. The contribution to H(,‘$‘) for one electron is then I$’Mi; l) = mu m’%
.
1 March 1985
CHEMICAL PHYSICS LETTERS
Volume 114, number 3
(18)
The interaction is the cross term in the combined kinetic energy for the two velocity fields. For a ground state, a first-order equation implies that the electron density will tend to increase where &de’) is negative and decrease where it is positive. It is positive where II,,, and u, have the same direction and negative where they are opposed. The velocity, u, depends on the phase and amplitude of the vibration. It is largest when the nuclei are moving at maximum speed. At these times the nuclei are at their equilibrium positions. The charge distortion that arises from H’del ) is then 90” out of phase with the usual charge distortion that arises during a vibration. The theory can now be applied to the simple case, the elu ring distortion mode of benzene. The situation at maximum distortion is symbolized by the ring at the right in fig. la. The ring on the left represents the situation l/4 cycle earlier. At this stage the nuclei are in their equilibrium positions and the usual dipole
t t H
>HYN\ H, Fig. 1. Diagrams describing MVCD (a) for the qu ring distortion mode of benzene and (b) for the e stretching mode of ammonia. The solid arrows represent uU, the electronic velocity induced by nuclear motion. The dashed arrows represent II,, the electronic velocity induced by the magnetic field. The circled + and - signs can be considered to indicate, respectively, that U, and 9, are parallel or antiparallel. They also can be considered to indicate the sipn of the induced second-order charge. The diagrams on the left represent the vibrations at the beginnings of their cycles (maximum speed, minimum displacement). The diagrams on the right represent the same vibrations l/4 cycle later (minimum speed, maximum displacement). The + and - siens describe the change in charge distribution in poing from the equilibrium configuration to maximum displacement. The solid continuous lines symbolize the contours of constant probability density.
vanishes. In order to form the dipole on the right, there must be a flow of electrons toward the bottom of the ring. The velocity field for this flow isu,. The electronic motion induced by the magnetic field circulates around the ring and is given by U, On the left side of the first ring U, and U, are parallel and mum -u, is positive; on the right they are an tiparallel and mu, -II” is negative. There is then an excess of charge (deficiency of electrons) on the left and a deficiency of charge on the right. This situation results in a weak transverse dipole followed l/4 cycle later by a strong vertical dipole. A rotating dipole is associated with circular dichroism, counterclockwise with left or positive circular dichroism and clockwise with right or negative circular dichroism. Fig. 1a shows an excess of clockwise flow and predicts negative circular dichroism in agreement with experiment It is possible to make some rough calculations of 267
Volume 114, number 3
CHEMICAL PHYSICS LETTERS
the circular dichroism for this vibration. The key is the fact that by using only the equation of continuity and the vibrational frequency, the current, and hence U, can be completely determined from ap/aQ. The vector u, can easily be obtained from the Larmor precession frequency. One can assume a one-electron interaction for H#c’) and solve eq. (17). In this case the six n-electrons of the ring are assumed to be the source of all the effect. and they are treated as particles on a circular track of radius Y. In the appendix the specific circular dichroism, B,/D,, is shown to be Be/Do = 64~r~rn~c~i%~/9i?~ ,
(19)
where V is the frequency in cm- 1 and the units of Be/Do are l/cm-l . The value of r is chosen such that the magnitude of the vector potential for the circular track is the same as that on a track that follows the carbon-carbon bonds in the ring, i.e.
s
A-ds=BA,,
(20)
where A, is the area of the ring. For benzene this yields 6&I\ = 31i2B 12/4
w>
(23)
On using 1= 1.39 A and V = 1480 cm-l it is predicted that B,/D, = -1.5 X lop5 per cm-l. The observed value is - 1.2 X 10-j per cm-l . The closeness of agreement is fortuitous; satisfactory agreement would lie within a factor of 2. A second example is the e stretching mode of NH,. In this case the nuclei contribute to the dipole moment, and the experimentally measured sign of ap/aQ [ 1 l] must be used in the theory. The dipole at maximum distortion is indicated on the right of fig. lb. The situation at the beginning of a vibrational cycle is shown on the left of fig. lb. The electrons tend to follow the nuclei and the magnetically induced electronic motion tends to follow contour lines of electron density. The induced out-of-phase dipole is predicted to point to the right, leading to 268
.
The quantitative expression for MVCD is (Aleft and the effect tends to be large when - Ari&%” A,, is small. For NH,, A,, is small. For NH,, A,, happens to be relatively small yielding a moderately large dichroism. A large value of A,, for the observed modes of paradichlorobenzene [3] provides an explanation other than symmetry for the fact MVCD is not observed for this compound. The prediction of strong symmetry dependence found in previous theory [4-61 does not appear in the theory given in this paper. At this point we return to the second term in H(d$ ) (cf. eq. (16~)). It can be thought of as compensating for the first term. Consider. as an example, an argon atom translating perpendicular to a uniform magnetic field. If the analysis that was done in fig. la for the benzene ring is applied to this case (an atom, including its electrons. moving down) a transverse dipole moment would be predicted, contrary to fact. For this system the appropriate gage function that keeps the vector potential centered on the nucleus of the atom (at R) is I’= -A-R
giving r= 3I121/2.
left circular dichroism. again in agreement with experiment
(21)
and for the ring 27rrlA 1= Brrr2 ,
1 March 1985
.
(24)
The second term is then a translational correction term and in this case it is readily apparent that it cancels the first term. It should be noted that in the treatment of the benzene molecule the gage function vanished everywhere, and only the first term contributed. Another problem to be considered is the vanishing of @dcu’ that was assumed in the development of eq. (17). This problem has been extensively investigated [ 10,121. In most cases, it is not possible to adjust the gage function so that dde”)$e vanishes everywhere. Fortunately the currents that are best approximated by using gage functions are those in the outer parts of molecules, regions where the largest contributions to MVCD are made. In principle the same considerations should be valid for the vanishing of die1 ) $e on adjusting II. In practice, for most vibrations non-vanishing circulation, which is an essential part of the treatment of magnetic effects, is only a minor consideration. It should then be expected that choosing u so that
1 March 1985
CHEMICALPHYSICS LETTERS
Volume 114, number 3
The dipole moment operator associated with the absorption of left circularly polarized radiation is
d$el ) $, nearly vanished would be a good approximation. This approach is consistent with that of using the equation of continuity (cf. eq. (27)) as was done in the treatment of the etu mode of benzene.
I*x+iy = -er(cos
Appendix
The usual procedure in treating vibrational absorption is to average the dipole moment over the electronic coordinates giving
The oscillating electronic (angular) density for one of the pairs of degenerate e,ti vibrations is pi = (6/27i) [I t ir sin 4 sin(wt)]
px+iJ> =
(25)
The velocity u, is u, = ViPi
I
(26)
where j is the current obtained tion of continuity,
by integrating
aj/aG =-$+/at. The quantity
the equa-
(27) u, is found to be
(35)
(36)
s PPx+iy d7el )
where p is the total electronic probability density. The part of p which makes a significant contribution to the circular dichroism is p=p~o~~p~l~+p~2~+p~3~=pj+p~2~,
(37)
where
t (#’
+ $(11’1))2 t (I@’
t @1))2]
+ 0 ((uB)~) .
u, = --aor sin @Icos(wt) t 0 (0”) The velocity u, is obtained sion frequency wL, i.e.
f$ + i sin 4)
(28)
from the Larmor preces-
(38)
From eqs. (25) (31), (34a, b, c), and (38) pc2) = 2B’sin @
(39)
and urn = WLY= eBr/2mc The Hamiltonian
The unperturbed
(30)
wavefunctions
are
I$~“) = n-l /2 sin $J,
cos f$
(31)
eqs. (17), (IS), (28) (29), and (30) yields
a2$,(1~1)/@2
+nf$i(L,L)
= _B’sin @ ruin) ,
(32)
where B’ = -amweBr The second-order
cos(ot)/fi2c
(33)
(40)
&+iv = -3aer [sin(wt) + 2iB/9] . In terms of the normal coordinate jugate momentum P, _ px+iy = -3Cer(Q + 2iDP/9w) ,
(41) Q and its con-
(42)
where CQ = a sin(wt) ,
(43)
CQ = CP = aw cos(wt) ,
(44)
D = B’/a cos(wt) ,
(45)
solutions of eq. (32) are
$b’>‘) = (27r)lj2 B' sin 4, 1//y)
p(l) = 6a cos 4 sin(wt) , yielding
a’/aQ
$60) = (271)-l i2,
Combining
(29)
operator for a particle on a ring is
tico) = -(fi2/2mrz)
$JJ’ = .-I/2
.
= d4B’/6)
li/$‘sl) = 7F1’2(~‘/6)
(-cos sin 24
(34b)
and C is a constant. For px_iy the sign of the second term would be negative. The transition moment between two neighboring vibrational states is found to be
(34c)
I%+
Pa) 24 - 3) )
,P@u dr, = s%c,
[-3erC(1
+ 2D/9)Q] av dr,, (46) 269
Volume
114. number
CHEMICAL
3
when the expression (47) is employed.
Since Aleft
is proportional
I(u + 11Eli,+izlu)2 and Arieht LUIA,,
to
to I(u + 1 IP~,i,lu)l*
= 4 X 2019 = 8rnweBr4/9A2c
To obtain
B,/D,
in I/cx-~
by eB/47rnzc2 yielding
it is necessary
a (48)
to divide
eq. (19).
References [ 1 ] T.A. Keiderling. J. Chem. Phys. 75 (1981) 3639. [2] T.R. Devine and T.A. Keiderlinf. J. Chem. Phys. 79 (1983) 5796.
270
PHYSICS
LETTERS
1 March 1985
[3] T.R. Devinc and T.A. Keiderling. J. Phys. Chem. 88 (1984) 390. [4] S.L. Cunningham and J.R. Hardy, Solid State Commun. 6 (1968) 769. 15 1 S. Datta and I’.S. Richardson, Intern. J. Quantum Chcm. 11 (I 977) 525. [6] L. Laus. V. Pultz. C. Marcott. J. Ovcrend and A. Mosco\vit/, J. Chem. Phys. 78 (1983) 4096. [7] T.H. Walnut and L.A. Nafie. J. Chem. Phys. 67 (1977) 1491. [8] S.T. Flpstein, .I. Chem. Phys. 58 (1973) 1592. 191 J. Overend, in: Vibrational intensities in infrared and Raman spectroscopy. eds. W.H. Person and G. Zerbi (Elsevier. Amsterdam, 1982) pp. 190-~202. [ 101 R.A. llepstrom and W.N. Lipscomb, Rev. Mod. Phys. 40 (1968) 354. [ 111 S. Abbate, S.L. Wunder and G. Zerbi, .I. Phys. Chem. 88 (1984) 600. [12] T.H. Walnut. J. Chem. Phys. 65 (1976) 3647.