Interpretation of Magnetic Hyperfine Coupling Constants The magnetic hyperfme structures of diatomic and linear free radicals are characterized Foley” hyperfme parameters (1, 2) which are usually defined as a = 2g,g0pfi_(
by the “Frosch-
h = ggIMFN((8?r/3)6(r)
- (3 cos”8 - 1)/2r’),,,
f = (3/2)gg,~~~((3
cos% - 1 )/r3&.
d = (3/2)X~,ro~~(sin’B/r~)~,.
The quantities ( )aV in the definitions above are to be averaged over the electronic space coordinates for the particular states. In most cases, contributions from the inner shells cancel out. and the average in the expressions ( I)-( 4) taken only over unpaired electrons is considered to be a good approximation. However, the hypetline coupling constants are sensitive to higher-order effects such as the exchange correlation, and occasionally experimentally determined hyperfine constants reveal discrepancies which cannot be explained by the simplified single configuration picture. For example, with this approximation. (h(r)),, and (sin%/ r’),, should always be positive or zero, and the relationship U=c’+d 3
should hold in a very good approximation. In the ground state of OH, (a(r)),+ is negative (3). Also recently, (sin28/r3),, was found to be negative in the ground state of BOZ (4). These negative values cannot be understood on the basis of the simplified interpretation. Even in the ground state of NO which is a good example of a simplified model. the hypertine coupling constant a is 84 MHz, while c/3 + d = 93 MHz (5-8). At the CI (Configuration Interaction ) level of calculations, part of the exchange correlation correction or the “spin-polarization” of the inner closed shells is taken into account automatically. However, expressions like those given as Eqs. ( I)-( 4) do not reflect the importance of the inner core polarization. More rigorous definitions of the average were given by several authors mostly in conjunction with ab initio calculations of hyperfine coupling constants (9. IO). The purpose of this Note is to address this classical but often forgotten problem to help understand that the effective values of (sin%/r’),, as well as (6(r)&,, can be negative. The magnetic hyperfme interaction terms have three origins: ( 1) the electron-orbital and nuclear-spin interaction, (2) Fermi contact interaction, and ( 3) the electron-spin and nuclear-spin dipolar interaction. They are written as follows. in terms of spherical tensor operators ( 11, 12): 1. Electron-orbital
and nuclear-spin ,&
= ?g,!.&+*, x c (-l)qr:“(I).
I 0 2. Fermi contact
X Z C-l)“~~‘)(I).
1 4 3. Electron-spin ‘&
TJ”(s,)+ TjLi(I)* C$)(B,,,$,,)/ri,.
NOTES Here Cr’( 8, 9) is defined in terms of the spherical
Cb”‘(@,, @,) = [4~/(2n
Now we consider only AS = 0 matrix elements because the second-order effect of AS # 0 elements on the hyperfme structure is mostly negligible. The matrix element of T$“‘(s,) is given. using the Wigner-Eckart theorem ( 12), as
using this relationship.
the effective hyperfine
H,i, = HI, + &
is given as
HIL = C (-l)qT;‘)(I).aI’,’
HFc = &I. S
P - Y
HD = (3O)‘“c p.y 4 where the coupling
in the effective Hamiltonian
2qlbLoILNC (Ii +
hF = (h/3)gg,p,p,
given above are defined by
f 16) l~SSj/S,
(18) The HI, term defined by Eq. ( 13) has AA = 0 and f I matrix elements. can be neglected in most cases. and HIL can be simply written as
The ALL = f 1 interaction,
H,, = al,‘L1,
The b and c constants
defined by Frosch and Foley
c = 3&’
( 1) are related to the parameters
given above as
cos%, - l)/rQASS)/S,
(22) The dipolar interaction has AA = + 1 and Aa = +-2 matrix elements in addition to those of ALI = 0. Among the off-diagonal matrix elements, only the AA = +2 matrix elements in *H states are significant except in cases of accidental degeneracy. They give rise to the “hyperfine doubling” in ‘II states (I, 13). and the d constant is defined as d = (3/2)gg,p0pN
2 (:’ = 1, SSls,,.sin’8,/r:.e”~
III = -1. 5X)/S
2 (,Z = ~1, SSl.~,~.sin’B,/r:.e-“*
/li = 1. SS)/S.
In the construction of molecular wavefunctions. two electrons with opposite spin components are allowed to occupy the same orbitals. As a result, the probability for the two electrons with opposite spin components to occupy the same spatial position is larger than it should be. Configuration interactions rectify this shortcoming. Or the unrestricted Hat-tree-Fock (UHF) method which assumes different orbitals for different spin components will take this exchange correlation effect into account. The spin-dependent averages which appear in the coupling constants given above can be taken separately over the a-spin and &spin rather than over unpaired electrons and the closed shells: (C
B,))s,=s = [(P(r.
Here we use an abbreviated notation. (P(r, S))* or (P( r, /I))#_ More rigorously, these can be expressed in terms of the density matrix (see for example Ref. ( 14)). We present a simple three-electron system as an example. We assume the total wavefunction to be given as
&))s~=.Y = (P(r,
(P(r. fl)),, =
The meaning of Eq. (25) is obvious. If the molecular orbital& 4, and &, are equal, as usually assumed to form a closed shell, the average can be taken only over the unpaired electrons. However. in general. bI is not equal to &. Therefore, the inner shell polarization may not be ignored particularly when the contribution from the unpaired electrons is small. In OH. the unpaired electron occupies the lrr- orbital which localizes at the 0 atom. With a single configuration approximation, (Z, 2.5,; 6(r,,)),=s = 0. and the hyperfine constants due to the proton arc greatly affected by the inner shell polarization. resulting in the negative spin density (6(r)),,, although ( sin%/r3),, is found to be positive (.3. 10). Recently. it was found that d of BOz in the ground state (d*B,) is negative (4). The ground electron configuration of BOz is * * . .(4~~)~(30,,)‘( IT,)~( IT,~)~. The l?r, orbital essentially localizes on the 0 atoms. Therefore, the hyperfine constants of BOz due to the B nucleus arc relatively small. If the contribution from the unpaired electrons is small. then the polarization of the inner shell due to the exchange correlation becomes important. Since the unpaired electron with the
It is interesting
to note that we can obtain (C
from electric quadrupole = [(&r,
H)),, + (p(r.
Therefore it is possible to obtain (P( r. 0)), and (P( r, H)), separately from the magnetic hyperfine constants and the electric quadrupole interaction constants. In ‘Il states, the sign of d, the sign of the h-doubling constant fi (or p + 2q). and the parity of the levels are closely connected. If the parities of the levels are determined by analyzing, for example, a *8 +-211 or *Z--*n band. then the sign [email protected]
uniquely determined and so is the sign ofd. However. it is often impossible to uniquely assign the parity (e- or jisymmetry) of the levels. and in such cases the sign of fi has to be assumed based on the pure precession assumption which is very poor. The sign of (C, 2s,,.sin2Bd/r5),~,~ is positive in most cases. BOz is an exceptional case. It is relatively easily concluded whether the proposition that ( 1, 2s,:. sin28,,/r:,)SZ=S is positive is safe by examination of the electron configuration. The unique assignment of the parities of the levels based on the sign of d is possible as demonstrated for NS ( IS). ACKNOWLEDGMENT The author
J. K. G. Watson
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Herzhvrg Institute ~~f’Asrrophysics National Research Council Oituwa. Onturio. Canada Kl.4 OR6 Received A uxust 7. 1990