The hypermagnetizability of molecular oxygen
Dan Jonsson, Patrick Norman, Olav Vahtras, and Hans Ågren
Department of Physics and Measurement Technology, Linköping University, S-58183, Linköping, Sweden
Antonio Rizzo
Istituto di Chimica Quantistica ed Energetica Molecolare, Consiglio Nazionale delle Ricerche,
Via Risorgimento 35, I-56126, Pisa, Italy
~Received 19 December 1996; accepted 20 February 1997!
The hypermagnetizability and the hypermagnetizability anisotropy of the oxygen molecule are
computed using cubic response theory applied to multi-configurational self-consistent field wave
functions. The effects of basis set, electron correlation, frequency dispersion, zero point vibrational
averaging and pure vibrational contributions are discussed. The result for the anisotropy
(D h 512.65 a.u. at l 5 632.8 nm!, even taking into account possible limitations in the treatment
of electron correlation and in the incompleteness of the basis set, maintains a different sign and is
more than two orders of magnitude smaller than the experimental values published in the literature.
Possible reasons for this large discrepancy are discussed. © 1997 American Institute of Physics.
@S0021-9606~97!01720-0#
I. INTRODUCTION
Calculations of high order electric, optical and magnetic
properties of molecules in the gas phase have in recent years
become more and more abundant due to the remarkable advances in both theoretical methods and computational techniques available to quantum chemists. Among these properties, the mixed electric and magnetic hyperpolarizabilities,
e.g., the hypermagnetizabilities and the corresponding
anisotropies, have received a great deal of interest due to the
close relationship they hold with the Cotton–Mouton Effect
~CME!,1 which describes the birefringence induced by a
magnetic field perpendicular to the direction of propagation
of the radiation beam. Although the first evidence of the
effect dates to the early days of the century, the topic has
received special attention in recent years because of the connections existing between the ‘‘classic’’ Cotton–Mouton effect observed in atoms and molecules and the prediction in
quantum electrodynamics of the existence of magnetic field
induced birefringence in vacuum, due to vacuum
fluctuations.1
Molecular oxygen plays an important role in the experimental study of CME of gases. Measurements of the
Cotton–Mouton constant of O2 2–4 show that the effect is
quite strong, and an order of magnitude larger than that observed in molecular nitrogen.1 If N2 has in the past been used
as a reference gas for the determination of the CME of other
gaseous samples,5–7 O2 might be a likely contaminant in
some specimens, and the accurate determination of its molar
Cotton–Mouton constant and of its hypermagnetizability anisotropy is therefore of great relevance.
Experimental evidence has been given of the fact that
the strength of the effect in O2 stems from the paramagnetism of the system.3,4 Kling et al.3,8 studied the theoretical
aspects connected with the presence of a non-vanishing magnetic dipole moment in a molecular system, and its implications for the Cotton–Mouton experiments. Kling, Dreier and
Hüttner measured the CME of molecular oxygen in the 200
8552
J. Chem. Phys. 106 (20), 22 May 1997
K to 400 K temperature range ~and at the liquid-nitrogen
temperature!, using a radiation beam with a wavelength of
632.8 nm.3 Their data are in good agreement with those of
Carusotto et al.,2 who employed a 514.5 nm radiation beam,
and were confirmed by Lukins and Ritchie a few years later.4
A full quantum mechanical derivation of the Cotton–Mouton
constant of a paramagnetic molecule is given in Refs. 3,8,
although general expressions ~not restricted to diamagnetic
systems! were also discussed in earlier works ~see Refs.
9,10!. In the ‘‘high’’ temperature limit, a diamagnetic and a
paramagnetic contribution to the overall constant can be
singled out. The diamagnetic contribution, which corresponds to the ‘‘semiclassical’’ Cotton–Mouton constant introduced by Buckingham and Pople,11 was estimated by subtracting the paramagnetic computed contribution from the
experimental data at each temperature by both Kling et al.3
and Lukins and Ritchie.4 Finally, the hypermagnetizability
anisotropy, at l5 632.8 nm, was obtained by an extrapolation to infinite temperature. The resulting values are
2737.6401. a.u.3 and 2603.6469. a.u.,4 respectively. Note
that, in spite of the wide error bars, representing doubled
standard deviations, these anisotropies are exceptionally
large when compared to those measured and/or computed for
other molecules of comparable structure and size; cf. N2
@ D h 5 27.9 a.u.,12 24.5 a.u.,13 97.675. a.u. ~exp.!14#,
C2 H2 @ D h 5 86.9 a.u.,12 20.654. a.u. ~exp.!,15 455.634 a.u.
~exp.!16#, HCN (D h 5 41.4 a.u.12!, CO @ D h 5 35.4 a.u.,17
37.9 a.u.,13 7.56 59.7 a.u. ~exp.!14# or HF (D h 5 6.6 a.u.,18
8.6 a.u.19!.
We have taken a great interest in recent years in the
study of electric field effects on electric and magnetic properties. The CME, and in particular the hypermagnetizability
anisotropies, of several atomic20–22 and molecular12,17,18,23,24
systems in the gas phase were studied by employing the
MCSCF response, either linear in connection with finite field
techniques or, more recently, by the cubic response. In the
latter case, the frequency dependence of the observable could
0021-9606/97/106(20)/8552/12/$10.00
© 1997 American Institute of Physics
Jonsson et al.: Hypermagnetizability of molecular oxygen
be studied. Very recently, our techniques were extended to
the study of the CME of liquid water.25
In this paper we analyze the oxygen molecule in the gas
phase. We obtain the hypermagnetizability tensor and its anisotropy of this molecule by employing a MCSCF cubic response approach. A study of the convergence properties with
respect to the basis set extension and electron correlation is
performed. The dependence on the choice of the magnetic
gauge origin is analyzed, as are the effects of frequency dispersion and of the zero point vibrational averaging. An estimate is also given of the residual pure vibrational contribution to the hypermagnetizability. Our best result for the
anisotropy of the hypermagnetizability at a wavelength of
632.8 nm is 12.65 a.u., i.e. with opposite sign and two orders of magnitude smaller ~in absolute value! than the experiment. The pure vibrational contribution alone reduces the
zero-point vibrationally averaged value ~16.57 a.u.! by
23.91 a.u. The discrepancy with respect to experiment is
thus huge.
Following a section on the theory, the computational details will be given as well as an account of the steps leading
to the estimate of the hypermagnetizability anisotropy. We
then compute the diamagnetic contribution to the Cotton–
Mouton constant and venture in a prediction of the total
CME of molecular oxygen. The possible causes of the disagreement with experiment are discussed at the end.
A. The Cotton–Mouton effect of paramagnetic
molecules
The description of the CME in paramagnetic molecules
requires a fully quantum mechanical treatment allowing for
proper account of the interaction between the electronic spin
and the molecular frame. This was accomplished by Kling
et al.3 in 1983 ~see also Refs. 8–10!, with an application to
the CME of molecular oxygen3 and nitric oxide8 in the gas
phase. In the limit of infinite temperature T, and neglecting
the dependence on the angular frequency of the electric field,
Kling et al.3 expressed the Cotton–Mouton constant m C 11 of
a molecule as
dia
m C T→` 5
HS
S
H S
S
D
DJ
~1!
2pNA 1
1
h ab , ab 2 h aa , bb
27 5
3
1
1
1
a j 2 a j
5kT ab ab 3 aa bb
~2!
,
1
2pNA
1
para
m C T→` 5
2 2 a ab m a m b 2 a aa m b m b
27 5k T
3
2
1
1
z
m 2 z
m
5kT ab , a b 3 aa , b b
DJ
,
a the electric dipole polarizability, and j the magnetizability.
z and h are the first and second hypermagnetizabilities, respectively, defined as
z ab , g 5
] a ab
,
]Bg
h ab , g d 5
~4!
] 2 a ab
] 2j gd
5
,
] B g] B d ] F a] F b
~5!
where B indicates the magnetic induction field and F the
electric field. In a homonuclear diatomic molecule symmetry
reduces Eqs. ~2! and ~3! to
dia
m C T→` 5
5
para
m C T→` 5
S
2pNA
2
DaDj
Dh1
27
15kT
2pNA
@ D h 1Q ~ T !# ,
27
S
D
~6!
D
2
2pNA
Dm 2 D a .
27 15k 2 T 2
~7!
The anisotropies in Eqs. ~6! and ~7! are defined as
D a 5 a zz 2 a xx , D j 5 j zz 2 j xx , Dm 2 5m 2z 2m 2x and
Dh5
1
~ 7 h xxxx 25 h xxy y 22 h xxzz 112h xzxz 22 h zzxx
15
12 h zzzz ! ,
~8!
with z as the internuclear axis. The temperature dependent,
electron spin related, molar magnetizability of a homonuclear paramagnetic molecule, j spin , can be written as
(m' m B gS) 26,27
II. DEFINITIONS AND METHOD
dia
para
m C T→` 5 m C T→` 1 m C T→` ,
8553
D
j spin
zz '
j spin
xx '
S ~ S11 ! N A m 2B g 2z
3kT
S ~ S11 ! N A m 2B g 2x
3kT
,
~9!
,
~10!
so that Eq. ~7! for the case of O2 can be approximated with
para
m C T→` 5
'
S
2p 1
D j spin D a
27 5kT
S
~11!
In the equations above, m B is the Bohr magneton and g
5 $ g x ,g y ,g z % is the electron spin g factor tensor (g x 5g y in
O2 ).
Finally,
S
4.2103103104
DaDj
T
5A @ D h 1Q ~ T !# ,
where the Einstein summation over repeated indices is implied. In these equations N A denotes Avogadro’s number, k
is the Boltzmann’s constant, m the magnetic dipole moment,
D
2 p N A 2 m 2B
@ g 2 2g 2x # D a .
27 15k 2 T 2 z
dia
m C T→` 5A D h 1
~3!
D
para
m C T→` 'A
S
D
D
3.3237573109 2
@ g z 2g 2x # D a ,
T2
~12!
~13!
when the temperature is given in Kelvin and the anisotropies in atomic units.1,28 A53.758748310221 if m C is
J. Chem. Phys., Vol. 106, No. 20, 22 May 1997
Jonsson et al.: Hypermagnetizability of molecular oxygen
8554
given in units of cm3 G22 mol21 ~4p e 0 ), while
A55.935561310231 if m C is given in units of m5 A22
mol21 . 1
An analysis of the limits of validity of Eq. ~1! at finite
temperatures was made in Ref. 3. The authors state that the
equation ‘‘represents an excellent approximation for all practicable temperatures’’ for molecules with reasonably spaced
rotational levels. They observe very small deviations
(,1%! from the general ~and much more complex!
expression—given in their paper—at temperatures well below room temperature.
The analysis of the experimental data for m C T→` , taken
at varying temperatures, allows in principle the determination of the hypermagnetizability anisotropy and of the products D a D j and Dm 2 D a , for instance by a simple fit with a
polynomial in powers of T 21 . If one of the four quantities
(D h ,D a ,D j and Dm 2 ) is known with sufficient accuracy
from other sources, the remaining three could in principle be
extracted from the Cotton–Mouton experiment. Both Kling
para
et al.3 and Lukins and Ritchie4 employed a m CT→`
, Eq. ~7!,
which was computed from the zero-field wave functions, the
zero-field energies and from the electronic and rotational
spectra of the oxygen molecule,29–31,27 and estimated D h
dia
para
from m C T→`
5 m C T→` 2 m C T→`
.
B. Hypermagnetizability tensors and response theory
To define the tensor h one can consider the double perturbation to the zero-field Hamiltonian in a static electric and
magnetic field,32
1
V52 m •F2m•B2 B• ĵ d •B,
2
~14!
where
m a 52 u e u ( r i,Ga
~15!
i
is the a component of the electric dipole moment operator
and r i,Ga 5(r i, a 2R G, a ) is the electronic coordinate of electron i with respect to the gauge origin (G);
m a 52
ueu
2m e
(i ~ l i,Ga 12s i, a !
~16!
is the magnetic dipole operator, involving the angular momentum operator l i,Ga 5 e abg r i,Gb p i, g —e abg being a third-rank
alternating tensor, p i, g the g component of the linear momentum operator for electron i—and s i, a denotes the a component of the electron spin operator for electron i;
d
ĵ ab
52
e2
4m e
(i @~ r i 2R G ! 2 d ab 2r i,Ga r i,Gb #
~17!
is the diamagnetic magnetizability operator. The elements of
the hypermagnetizability tensor in the Cotton–Mouton experiment, involving a static magnetic induction field, can be
obtained by a perturbative expansion of Eq. ~14!. They can
be shown to relate to quadratic and cubic response
functions33 as follows:
para
h ab , g d ~ 2 v ; v ,0,0! 5 h ab
, g d ~ 2 v ; v ,0,0 !
dia
1 h ab
, g d ~ 2 v ; v ,0,0 ! ,
para
h ab
, g d ~ 2 v ; v ,0,0 ! } ^^ r a ;r b ,m g ,m d && 2 v ; v ,0,0
1
52 ^^ r a ;r b , ~ l g 12s g ! ,
4
~ l d 12s d ! && 2 v ; v ,0,0 ,
~18!
1
52 ^^ r a ;r b ,l g ,l d && 2 v ; v ,0,0 , ~19!
4
1
4
dia
2
h ab
, g d ~ 2 v ; v ,0,0 ! 52 ^^ r a ;r b , ~ r d g , d
2r g r d ! && 2 v ; v ,0 ,
~20!
where v is the circular frequency of the propagating beam
and the dependence on the origin of the gauge has been
neglected for the sake of brevity. Electron spin disappears
going from Eq. ~18! to Eq. ~19! because the response function involves mixtures of pure spin and pure orbital
operators—it is generally true that the response function vanishes whenever one of the operators commutes with the
Hamiltonian as well as all the other operators. This is easily
seen in the time representation, where the linear response
function is the integral kernel ^ @ A(t),V(t 8 ) # & , 34 a commutator of the operators in the interaction representation. It is
clear that this response function vanishes if V commutes with
H 0 and A. Higher order response functions involve coupled
commutators of higher order,35 but the generalization of the
previous statement is straightforward.
The efficient implementations of multi-configurational
SCF ~MCSCF! response theory in various contexts36–38 have
recently been generalized to cubic response functions.18 The
coding involves the use of direct one-index transformations,
direct atomic orbital constructions of matrices and direct iterative linear transformations in the solutions of the response
equations. In this paper the MCSCF cubic response is employed to compute the hypermagnetizability anisotropy of
molecular oxygen in its ground 3 S 2
g electronic state.
C. Gauge origin dependence
In finite basis set calculations using the standard magnetic gauge-dependent approach, the magnetizability and the
hypermagnetizability depend on the choice of the magnetic
induction origin. Our current implementation of cubic response does not yet allow for the use of the so-called
London-Atomic Orbitals ~LAO’s, or Gauge Invariant Atomic
Orbitals, GIAO!39–43 as basis sets in the calculation of magnetic properties. LAO’s were employed in previous studies
of hypermagnetizabilities and CME performed employing a
linear response plus finite field approach,12,17,23 and were
shown to enhance the basis set convergence rate of the properties. However, the O2 molecule is sufficiently small to allow for applications with large basis sets without LAO’s.
Cybulski and Bishop13 discussed the magnetic gauge dependence of the magnetizability and hypermagnetizability in
J. Chem. Phys., Vol. 106, No. 20, 22 May 1997
Jonsson et al.: Hypermagnetizability of molecular oxygen
homonuclear diatomic molecules. For a displacement of the
distance R, the hypermagnetizability changes according to
dia
dia
h ab
, g d ~ R ! 5 h ab , g d ,
~21!
para
para
h ab
, g d ~ R ! 5 h ab , g d 1
(
pr m n
mn
e g p m e d rn R p R r N ab
,
1
mn
N ab
5 ^^ r a ;r b , p m , p n && .
4
~22!
~23!
In a complete basis, where the results are gauge independent,
mn
mn
50. The magnitude of N ab
is therefore a measure of the
N ab
quality of the employed basis set in a given calculation. The
corresponding sum rule that is to be fulfilled for the gauge
dependent magnetizability @the temperature dependent part,
j spin , Eqs. ~9! and ~10!, is not important in this context#,
1
para
dia
d
j ab 5 j ab
1 j ab
52 ^^ l a ;l b && 2 ^ 0 u ĵ ab
u0&,
4
~24!
is easily obtained by comparing the following equations:13
1
dia
dia
j aa
2 ne
~ R! 5 j aa
4
para
para
j aa
1
~ R! 5 j aa
1
4
(n ~ 12 d a n ! R 2n ,
~25!
(n ~ 12 d a n ! R 2n n nnP P ,
~26!
n aP nP 5 ^^ p a ; p n && ,
~27!
where n e denotes the number of electrons. In a complete
PP
basis n e d ab 5n ab
. In the following we will discuss the dependence of our results on the magnetic gauge origin using
the relationships introduced above.
Ref. 44. An account of the progress in the subject, with specific references to the CME, can also be found in Refs. 1,45.
Results obtained for the property X at the equilibrium
internuclear distance need to be corrected for zero-point vibrational effects. This is generally accomplished by computing the expectation value,
X̄5 ^ y ~ g ! u X u y ~ g ! & ,
`
U5U 0 2
F
n11
1
n!
X ~ab
( (
•••l ; m n ••• t
n50 i50 ~ n11 ! !
i
3F a F b •••F l i B m B n •••B t n112i ,
The effect of nuclear vibrations on the electric and magnetic properties and in particular on the polarizabilities, magnetizabilities and hypermagnetizabilities have been studied
in detail by Bishop and his collaborators; see for instance
52
i ; m n ••• t n112i
~28!
over the lowest vibrational state u y (g) & of the electronic
ground state. We note that Bishop et al.32 also included rotations in their study of the hypermagnetizability anisotropy
of H2 and D2 . As furthermore shown by Bishop,44 vibrations
influence electric and magnetic properties also through the
effect the radiation has on the vibrational motion, which
gives rise to the so called pure vibrational ~hyper!polarizabilities. The pure vibrational contribution to the hypermagnetizability of a diamagnetic molecule is discussed in Refs.
13,46.
The case of a paramagnetic molecule, e.g. with a permanent magnetic moment, has to our knowledge never been
studied. To obtain the vibrational contribution to the hypermagnetizability we follow Bishop.44,45,47 We concentrate on
the static pure vibrational contribution, which for many optical processes is larger than that appearing at finite external
field frequencies. The equations can easily be generalized to
the dynamic case, and we will discuss frequency dependence
at the end of this section. We use perturbation theory, and for
a molecule in static uniform electric F and magnetic induction B fields, we write the electronic energy U as44,45,47
D. The effect of molecular vibrations
n!
X ~ab
•••l
8555
G
F
] 4U
] F a] F b] B g] B d
G
F a 5F b 5B g 5B d 50
y
5 h ab
,gd .
~31!
~30!
.
F a 5F b 5•••5F l 5B m 5B n 5•••5B t
i
3!
X ~ab
; g d 52
~29!
where, for the purpose of introducing the general formulae,
we adopt a slightly extended version of the notation of
Bishop,48 suitable for combined electric and magnetic fields.
Thus in Eq. ~29! we have
] n11 U
] F a ] F b ••• ] F l i ] B m ] B n ••• ] B t n112i
A semicolon is used to separate ‘‘electric’’ ~left! from
‘‘magnetic’’ ~right! indices. Note that pure electric properties
(n)
are indicated as X ab
•••l i ; , pure magnetic properties as
(n)
X ; ab •••l . l i is the i-th letter and t n112i the (n112i)-th
i
letter of the Greek alphabet. Einstein summation over repeated indices is implied in Eq. ~29!. For instance,
n112i
n112i
50
The notation could be generalized to the case of nonhomogeneous fields by introducing commas ~neglected here for
the sake of simplicity! between indices. For instance, in the
presence of an electric field gradient (F bg ),
X ~a3, !bg ; d 52
F
] 4U
] F a ] F bg ] B d
G
.
~32!
F a 5F bg 5B d 50
If we take the perturbative expansion in Eq. ~29! through
fourth order and collect terms involving products of the type
J. Chem. Phys., Vol. 106, No. 20, 22 May 1997
Jonsson et al.: Hypermagnetizability of molecular oxygen
8556
F 2 B 2 , the general expression for the vibrational hypermagnetizabilities is obtained. The procedure has been described in detail
in Refs. 44,47. The result is
3!
y
X ~ab
; g d 5 h ab , g d 5
1
2
1!
~1!
@ 2 ~ X ~a1; !g ! g,k ~ X ~b1; !d ! k,g 1 ~ X ~a0; ! ! g,k ~ X ~b2; !g d ! k,g 1 ~ X ~b0; ! ! g,k ~ X ~a2; !g d ! k,g 1 ~ X ~ab
; ! g,k ~ X ; g d ! k,g
\ kÞg v k
(
2!
~0!
~2!
1 ~ X ~; g0 ! ! g,k ~ X ~ab
; d ! k,g 1 ~ X ; d ! g,k ~ X ab ; g ! k,g # 1
3 ~ X ~a0; ! ! g,k ~ X̄ ~; g0 ! ! k,l ~ X ~b1; !d ! l,g 1
1!
3 ~ X̄ ~; d0 ! ! k,l ~ X ~ab
; ! l,g 1
1
\2
1
\2
~8!#
1
1
(P kÞg
( lÞg
( v k v l ~ X ~; g1 d! ! g,k~ X̄ ~a0;! ! k,l~ X ~b0;! ! l,g
1
1
1
\3
(P kÞg
( lÞg
( mÞg
( v k v l v m ~ X ~a0;! ! g,k~ X̄ ~; g0 ! ! k,l~ X̄ ~; d0 ! ! l,m~ X ~b0;! ! m,g
2
1
\3
(P kÞg
( lÞg
( v k v 2l ~ X ~; g0 ! ! g,k~ X ~; d0 ! ! k,g~ X ~a0;! ! g,l~ X ~b0;! ! l,g .
1
2
~ a ab ! gk ~ j g d ! kg
.
\ kÞg
vk
(
~34!
The hypermagnetizability anisotropy becomes @see Eq.
D h y5
4
~ D a ! gk ~ D j ! kg
.
15\ kÞg
vk
(
~35!
In a homonuclear diatomic molecule with a permanent
magnetic moment, terms involving odd numbers of electric
field indices in Eq. ~33! vanish, as do those involving
X a(2); g d , and
y
h ab
, g d 51
1
1
2
~j ! ~a !
\ kÞg v k g d g,k ab k,g
(
2
\ 2 kÞg
1
(P kÞg
( lÞg
( v kv l
(P kÞg
( lÞg
( v k v l ~ X ~; g0 ! ! g,k
In Eq. ~33!, ( P indicates the six ~twenty-four! possible
permutations of the three ~four! component indices in the
third ~fourth! order terms; ( k indicates a sum over the excited vibrational states u y (k) & of the ground electronic state,
the
lowest
state
being
explicitly
excluded;
(Y ) mn 5 ^ y (m) u Y u y (n) & for a given property Y ;
Ȳ 5Y 2 ^ y (g) u Y u y (g) & and \ v n 5\ v n,g are the vibrational
transition energies.
In a homonuclear diatomic molecule with no permanent
electric and magnetic moment, the terms involving odd numbers of magnetic or electric field indices vanish in Eq. ~33!,
and we are left with the well known expression1,13 ~we revert
now to the usual notation!
y
h ab
,gd5
4
\2
1
( lÞg
( v k v l @~ m g ! g,k~ m̄ d ! k,l~ a ab ! l,g
1 ~ m g ! g,k ~ ā ab ! k,l ~ m d ! l,g
1 ~ a ab ! g,k ~ m̄ g ! k,l ~ m d ! l,g # .
~36!
~33!
Once again, employing Eq. ~8! it is easy to obtain
D h y5
4
~ D a ! gk ~ D j ! kg
15\ kÞg
vk
(
1
8
15\ 2 kÞg
( lÞg
(
~ m ! gk ~ m̄ ! kl ~ D a ! lg
.
v kv l
~37!
This equation was obtained independently by Bishop.49
The second term vanishes in diamagnetic molecules, and has
here been derived for paramagnetic molecules for the first
time. In the present case, since m does not depend on the
internuclear coordinate, its contribution also vanishes. In
principle the magnetic dipole moment function appearing in
Eq. ~37! does depend on the internuclear distance, the dependence being introduced by the terms neglected in our approximation m5 m B gS. On the other hand corrections due to
nuclear spin ~in molecules with 17O nuclei!, rotational
nuclear and electronic magnetic moments27,50 are small and
account for less than 0.1% of the discrepancies between experimental and computed positions of the lines of the fine
structure spectrum of O2 . 27 We thus believe that the actual
contribution of the second term in Eq. ~37! to the hypermagnetizability anisotropy of O2 could safely be considered
negligible.
The extension to the dynamic case can be obtained following Bishop and his collaborators.44,47,51,52 The result is
that the appropriate dynamical expression is still given by
Eq. ~37! if we just substitute D a (2 v ; v ) for D a (0;0).
Since the frequency dependence enters solely from D a it
follows that the vibrational contributions may be even more
important for optical frequencies than for the static case. This
is in contrast to electric hyperpolarizabilities where it has
been shown ~see for example Ref. 51! that the relative importance of vibrational hyperpolarizabilities decreases moving from the static to the dynamic case, although it might be
J. Chem. Phys., Vol. 106, No. 20, 22 May 1997
Jonsson et al.: Hypermagnetizability of molecular oxygen
8557
TABLE I. Basis set description.
Basis set
No. uncontracted
No. contracted
Contraction
Ref.
b1
b2
b3
b4
b5
h1
h2
35
58
44
74
90
70
79
23
46
32
62
78
67
76
@ 10s5 p2d/4s3 p2d #
@ 11s6 p3d2 f /5s4 p3d2 f #
@ 11s6 p3d/5s4 p3d #
@ 12s7 p4d3 f /6s5 p4d3 f #
@ 13s8 p5d4 f /7s6 p5d4 f #
@ 12s8 p4d2 f /9s8 p4d2 f #
@ 13s9 p5d2 f /10s9 p5d2 f #
aug-cc-pVDZ
aug-cc-pVTZ
daug-cc-pVDZ
daug-cc-pVTZ
taug-cc-pVTZ
HIVa
HIVb
quite significant for particular systems and selected optical
processes, especially when the possibility of near-resonances
arises.
III. RESULTS AND DISCUSSION
A. Computational details
Due to the well known extreme sensitivity of the hypermagnetizability calculations on the choice of basis set, and to
the fact that LAO’s are currently unavailable with our methods, which would greatly enhance the convergence rate with
the extension of the basis, calculations were performed with
many sets and the convergence with the extension and quality of those basis sets was closely studied. Table I summarizes the characteristics of the different basis sets employed.
Basis sets b1 to b5 are part of the correlation consistent set of
Dunning and Woon53,54 especially designed for electric perturbations and which were found particularly suitable to
compute magnetizabilities55,56 and hypermagnetizabilities.25
Basis sets h1 and h2 were introduced in Ref. 57 and employed in several subsequent studies of electric field effects
on magnetic properties.12,17,23
Three different complete active space SCF ~CASSCF!
reference wave functions were employed in the multiconfigurational cubic response calculations. The number of
inactive and active orbitals is given in the following with
reference to the irreducible representations of the D2h group:
the notation n 1 n 2 •••n 8 indicates the number of orbitals in
the s g p ux p uy d g s u p gx p gy and d u symmetries respectively:
• CAS-1: inactive ~3 0 0 0 2 0 0 0!
active ~0 1 1 0 0 1 1 0!;
• CAS-2: inactive ~2 0 0 0 2 0 0 0!
active ~1 1 1 0 1 1 1 0!;
• CAS-3: inactive ~2 0 0 0 1 0 0 0!
active ~2 2 2 0 2 1 1 0!;
CAS-1 is a 1 p u 1 p g ~two determinants! complete active
space, the simplest multi-configurational wave function for
the system under study. With CAS-2 the 3 s g orbital enters
the active space. Our largest active space ~CAS-3! includes
the 2 s u , 3s g , 1p u , 1p g , 3s u , 4s g and 2 p u orbitals, and
with respect to the usually employed so called Full-Valence
CAS, it only excludes the 2 s g orbital—which is a truly ‘‘inactive’’ orbital—while including the 4 s g and the 2 p u orbitals.
The equilibrium distance for molecular oxygen was
taken to be 2.28167 a.u.58 All calculations were performed
using the DALTON program.59
B. Basis set extension and electron correlation
effects
Table II summarizes the results obtained at the equilibrium geometry for the static paramagnetic, diamagnetic and
total hypermagnetizability anisotropy of O2 , employing different basis sets and CASSCF reference wave functions. Results for the individual tensor components are available upon
request from the authors. The convergence with respect to
extension and characteristics of the basis set and with the
sophistication in the treatment of electron correlation is generally quite satisfactory. Deviations between the results obtained employing the CAS-2 or the CAS-3 reference wave
functions are small for most of the tensor components. Some
para
exceptions exist, h xz,xz
being perhaps the most evident one.
Its value goes from 25.7 a.u. ~CAS-2! to 28.3 a.u. ~CAS-3!,
a 30% variation. Due to the particular influence this tensor
component has on Eq. ~8!, the anisotropy of the hypermagnetizability is perturbed quite noticeably, being reduced in
TABLE II. The anisotropy of the hypermagnetizability for the O2 molecule. Static values. Atomic units.
Basis set
D h para
D h dia
Dh
D h para
D h dia
Dh
D h para
D h dia
Dh
b1
b2
b3
b4
b5
h1
h2
25.64
211.07
217.48
214.93
214.69
214.93
215.01
CAS-1
12.63
17.03
19.50
21.84
22.22
21.13
21.28
6.99
5.96
2.02
6.91
7.53
6.20
6.27
24.89
29.99
216.05
213.56
213.34
213.55
213.63
CAS-2
12.90
17.40
19.91
22.14
22.51
21.46
21.61
8.01
7.41
3.86
8.58
9.17
7.91
7.98
29.02
214.65
220.61
217.80
217.65
217.87
217.94
CAS-3
15.34
19.84
22.98
24.87
25.29
24.25
24.37
6.32
5.19
2.37
7.07
7.64
6.38
6.43
J. Chem. Phys., Vol. 106, No. 20, 22 May 1997
Jonsson et al.: Hypermagnetizability of molecular oxygen
8558
TABLE III. Sum rule results for gauge dependent ~hyper-!magnetizabilities.
Active space is CAS-3. Atomic units. The exact values in the last column
refer to a complete basis set.
xx
N xx
xx
Nyy
xx
N zz
yx
N yx
zx
N zx
zz
N xx
zz
N zz
PP
n xx
PP
n zz
TABLE IV. The dispersion of the anisotropy of the hypermagnetizability
for the O2 molecule. Active space is CAS-3 and basis set is h1. Atomic
units.
b1
h1
Exact
Wavelength~/nm!
D h para
D h dia
Dh
21.124
23.224
24.469
1.050
0.918
22.077
21.057
11.52
12.17
0.086
20.669
20.267
0.378
0.163
20.346
20.032
15.89
15.92
0
0
0
0
0
0
0
16
16
2278.
1064.
694.3
632.8
590.0
514.5
488.8
217.96
218.24
218.75
218.93
219.09
219.51
219.70
24.31
24.56
25.00
25.16
25.31
25.67
25.83
6.36
6.33
6.26
6.23
6.22
6.16
6.14
CAS-3 by 20–25% with respect to CAS-2. Correlation effects were found to be very strong in a recent calculation of
the hyperpolarizability of O2 , 60 especially for some components. The delicate balance between intra-valence and dynamic correlation was singled out to explain this finding.
Properties become quite stable as soon as basis sets of
triple zeta ~b4, b5! quality are employed. Also, results obtained with the h1 and h2 basis sets are similar, and in appreciable agreement with those of the correlation consistent
triple-zeta sets. In particular, once a CAS-3 reference wave
function is employed, the hypermagnetizability anisotropy
varies within the 16.38 a.u. to the 17.64 a.u. range upon
using the four sets b4, b5, h1 and h2, i.e. an overall 20%
variation ~see Table II!.
The computational requirements for the cubic response
method are quite demanding, and a delicate balance between
extension of basis set and of the active space in a CASSCF
calculation is needed. However, the conclusions of the
present study, as it will be soon evident, are totally unaffected by the degree of convergence shown by the hypermagnetizability anisotropy with respect to basis set extension or
electron correlation. To conjugate economy and accuracy,
we decided to adopt a CAS-3 reference wave function and
the h1 basis set as reference choices for the study of the
frequency dispersion, zero point vibrational average and vibrational contribution to the hypermagnetizability anisotropy. At the end we will come back to the problem of the
‘‘accuracy’’ of our results as dictated by the limitations in
basis set and active space expansions, and estimate a ‘‘confidence range’’ for our value of the hypermagnetizability anisotropy.
C. Gauge origin dependence
All results obtained in this work were obtained by adopting the center of nuclear masses as an origin of the magnetic
gauge. Since the current approach is not magnetic-gaugeorigin independent, an analysis of the extent to which the
results are unaffected by a change of gauge was performed.
The dependence of our magnetizability and hypermagnetizability tensor on the choice of the magnetic gauge origin can
be discussed by referring to Table III, where the values of the
sum rules introduced in Eqs. ~23! and ~27! are displayed. We
mn
show the values of N ab
and n aP nP for our reference basis set,
h1, and compare them with those obtained with our smallest
set, the aug-cc-pVDZ, @ 10s5p2d/4s3p2d # set labeled b1.
The reference wave function is the one labeled CAS-3.
PP
The sum rule for the magnetizability n aa
5n e 516 is
quite well satisfied in our reference set, with a mere 0.7%
deviation for the perpendicular ( a 5x) component and 0.5%
deviation for the parallel ( a 5z) component. The largest demn
viation from the sum rule N ab
50 is observed for the mixed
perpendicular component ( a 5 b 5y, m 5 n 5x). The smallxx
est deviations occur for N xx
zz and N xx . To give an idea of the
para
extent of the effect, h zz,zz would change from 0 a.u. to
N xx
zz 520.267 a.u. for a change in the magnetic gauge origin
from the center of mass of 1 a.u. along the x or y direction
@according to the results displayed in Table III and to Eq.
~22!#. With a displacement of 1 a.u. along the z direction the
para
h xx,y
y component would go from 137.34 a.u. to 136.67,
changing exactly by N xx
y y , a 1.8% variation. The other large
para
component, h zz,zz
, changes by less than 0.9% for the same
displacement. Larger variations can occasionally occur for
displacements along the x (y) directions by other components; for example, for a 1 a.u. displacement along x,
para
h xx,zz
changes by about 10% ~from 16.82 a.u. to 6.15 a.u.!.
The violations of the sum rules are much stronger for the
small b1 basis set.
D. Frequency dispersion
One of the advantages of the use of the cubic response
method over finite electric field approaches12,17,23 in hypermagnetizability calculations lies in the possibility of studying
the frequency dispersion of the response property, an essential feature when comparing to experiment, which is always
performed at finite laser frequencies. It has to be mentioned
that the frequency dependence of the hypermagnetizability
and hypermagnetizability anisotropies of N2 , H2 , HF and
CO could be obtained by Cybulski and Bishop by employing
a finite field magnetic technique coupled to MP2,13 MP319 or
L-CCD19 approaches. We also note that the frequency dispersion of the hypermagnetizability has been studied before
in atomic22 and molecular18,24 systems by cubic response calculations in our group.
Table IV shows the dispersion observed at the equilibrium geometry for the paramagnetic, diamagnetic and total
hypermagnetizability anisotropy of O2 . The frequency range
covers the regions of experimental interest and is far away
J. Chem. Phys., Vol. 106, No. 20, 22 May 1997
Jonsson et al.: Hypermagnetizability of molecular oxygen
TABLE V. The static hypermagnetizability anisotropy for the O2 molecule
for different internuclear distances R. Active space is CAS-3 and basis set is
h1. Atomic units.
R
D h para
D h dia
Dh
D h para
1.70
1.80
1.85
1.90
1.95
2.00
2.05
2.10
2.20
2.28
2.40
2.60
2.80
average
290.62
257.11
247.28
240.06
234.57
230.31
226.95
224.27
220.27
217.88
215.33
212.91
210.40
217.83
l5`
31.08
24.23
22.56
21.57
21.08
20.95
21.10
21.49
22.77
24.24
26.88
32.42
37.32
24.52
259.54
232.88
224.72
218.49
213.49
29.36
25.85
22.78
2.50
6.36
11.55
19.51
26.92
6.69
297.55
260.54
249.88
242.12
236.27
231.76
228.24
225.44
221.34
218.93
216.43
214.33
217.07
218.91
D h dia
Dh
l5632.8 nm
33.13
264.42
25.38
235.16
23.50
226.38
22.39
219.73
21.82
214.45
21.67
210.09
21.82
26.42
22.21
23.23
23.59
2.25
25.16
6.23
28.02
11.59
34.16
19.83
41.17
24.10
25.48
6.57
from the first excitation energy of the same multiplicity of
molecular oxygen ~lying at l,200 nm: A 3 S 1
u ' 6.3 4 6.5
eV; C 3 D u ' 6.0 4 6.2 eV61!. As indicated above, for this
study—and in the following—the largest CASSCF reference
wave function ~CAS-3! and basis set h1 were employed. A
very weak frequency dispersion is displayed by the anisotropy, which varies by 3.5% in the range of frequencies given
in Table IV. The individual components, and separately the
paramagnetic and the diamagnetic anisotropies, show a distinctly more pronounced frequency dependence, evidently
smoothed out by the ‘‘averaging’’ performed to obtain the
anisotropy, cf Eq. ~8!. The very weak dispersion of the hypermagnetizability anisotropy is common to most systems
studied to date; negligible dispersions are displayed by species like Ne,20,22 Ar,21 HF18 and liquid water.25
It should also be noted that our static paramagnetic contribution to the anisotropy of the hypermagnetizability
~217.87 a.u.; cf. Table II!, is in nice agreement with the
estimate obtained from the approximate Cauchy-type moment expansion1 ~a.u.!,
`
Dh
para
1
'2
v 2n ~ 2n11 !~ 2n12 ! S ~ 22n24 !
4 n50
(
1
'2 S ~ 24 ! ,
2
~38!
8559
which is rigorous for spherical symmetry and where S(2 j)
indicates the appropriate Cauchy moment ~sum rule!. In particular we are aware of an estimated value 2 S(24)/2
5217.37 a.u. obtained from integrated dipole oscillator
strengths by Zeiss et al.62,63
According to Table IV, the hypermagnetizability anisotropy of the oxygen molecule at l 5 632.8 nm is, within our
approximations and neglecting zero-point vibrational averaging and pure vibrational contributions, D h 5 16.23 a.u.
E. Zero-point vibrational averaging and pure
vibrational contributions
The effect of molecular vibrations on the hypermagnetizabilities and hypermagnetizability anisotropies has been
studied before for H2 , 64,32,13,19 D2 32 and N2 , HF and CO,13,19
see also Ref. 45. In particular, pure vibrational contributions
to selected tensor components were found to be particularly
important for N2 ~more than 50% of the electronic value, cf.
Ref. 13!.
Table V displays the dependence on the internuclear distance of the paramagnetic, diamagnetic and total hypermagnetizability anisotropy of O2 . Results in the static regime
and for l5632.8 nm are shown. The zero-point vibrationally
averaged hypermagnetizability anisotropy of molecular oxygen is D h 5 16.57 a.u. at a wavelength of 632.8 nm, using
our largest ~CAS-3! active space wave function and in basis
set h1. This corrects the equilibrium geometry value ~16.23
a.u.! by somewhat less than 6%.
Table VI shows the vibrational contribution to the anisotropy of the hypermagnetizability D h y at l5` and at l
5 632.8 nm, obtained by employing Eq. ~35!, i.e. by neglecting the second term appearing in Eq. ~37!. This is justified in
view of the arguments given at the end of Sec. II D. It is seen
that essentially only the first vibrational state contributes to
the overall value of D h y 5 23.57 a.u. (l5`) and D h y 5
23.91 a.u. (l 5 632.8 nm!. The second vibrational state
contributes in both cases to less than 0.1% of the overall
value. Notably, the effect is 9% larger at finite frequency
with respect to the static case; see the last paragraph of Sec.
II D. Also, the size of the effect is significant, correcting the
zero-point averaged anisotropy by about 60%.
Adding the pure vibrational contribution to the zeropoint averaged electronic contribution gives a D h of 12.65
a.u. at l5632.8 nm. The limitations of our calculations, discussed in Sec. III A, add error bars to this value, which
TABLE VI. The vibrational contribution to the anisotropy of the hypermagnetizability D h y (l 5 632.8 nm!.
Active space is CAS-3 and basis set is h1. The contribution of the first four vibrational states of the ground
electronic state is shown. Atomic units.
n
^ y (0) u D a (0;0) u y (n) &
^ y (0) u D a (2 v ; v ) u y (n) &
^ y (0) u D j u y (n) &
v 0,n
D h (0)
Dh(v)
1
2
3
4
20.6543
20.0195
0.0004
20.0002
20.7167
20.0136
20.0004
0.0002
0.1446
20.0123
0.0020
20.0000
0.0071
0.0140
0.0209
0.0276
23.5753
0.0046
0.0000
0.0000
23.9163
0.0032
20.0000
20.0000
23.5707
23.9131
Total
J. Chem. Phys., Vol. 106, No. 20, 22 May 1997
8560
Jonsson et al.: Hypermagnetizability of molecular oxygen
we estimate, perhaps even too pessimistically, to be at worst
61 a.u. This figure arises by summing the apparent error due
to missing basis set convergence (' 20%! to that due to a
possible inadequate description of electron correlation
(,25%!; cf. Sec. III A. Nonetheless, the calculated value
maintains a different sign and a much smaller absolute value
with respect to the experimental estimates, e.g., 2737.
6401. a.u.3 or 2603.6469. a.u.,4 respectively. As it will be
evident in the next sections, possible residual corrections to
our hypermagnetizability anisotropy are unimportant for the
determination of the diamagnetic contribution, and consequently for the magnitude of the CME of gaseous O2 .
F. The diamagnetic contribution to the Cotton–Mouton
constant of O2
The diamagnetic contribution to the Cotton–Mouton
constant can be estimated by employing Eq. ~6!. To this end
zero-point averaged anisotropies of the magnetizability and
of the electric dipole polarizabilities were computed, and the
dispersion of the latter was studied. The anisotropies entering
the temperature dependent term Q(T) in Eq. ~6! should include pure vibrational contributions in polar (D a ) and/or
paramagnetic (D j ) systems. In particular in our case the
pure vibrational contribution to the magnetizability anisotropy arising due to the existence of a permanent magnetic
dipole was neglected, the motivations being those explained
for the neglect of the magnetic dipole moment induced pure
vibrational contribution to the hypermagnetizability anisotropy; see Sec. II D.
The zero-point averaged temperature independent contribution to the magnetizability of O2 , computed with a CAS-3
wave function and basis set h1, is 22.44 a.u. By employing
Eqs. ~9! and ~10! with Evenson and Mizushima electron spin
g factors,30 see below, we estimate the spin-induced temperature dependent contribution to be j sapin
v e ' 1720. a.u.,
which gives j a v e ' 717.6 a.u., to be compared with an experimental value of 1726. a.u.65
The individual magnetizability (T independent! tensor
components are j xx 5 21.80 a.u. and j zz 5 23.71 a.u., with
an anisotropy D j 5 21.91 a.u. The anisotropy of the magnetizability of molecular oxygen was derived from the CME
experiment both by Kling et al.3 and by Lukins and Ritchie.4
Actually, both groups were able to determine from their experiments the product D a D j , and estimated D j by assuming
D a 57.36 a.u.66 Their value is D j 522.24 60.3 a.u.3 and
D j 522.7260.5 a.u.4 Lukins and Ritchie also cite a value of
D j 522.3360.2 a.u. obtained through Ramsey’s approximate formula.67 We are not aware of any electron correlated
calculations of the magnetizability anisotropy of O2 in the
current literature.
Table VII shows the dispersion of the electric dipole
polarizability of O2 for a range of wavelengths going from
` to 488.3 nm, and computed at the equilibrium internuclear
distance. For l5` and l 5 632.8 nm we also display the
zero-point vibrational averages. Frequency dependent electric dipole polarizabilities correlated calculations of the O2
molecule are present in the literature.68–70 In particular, the
TABLE VII. The dispersion of the electric dipole polarizability of the O2
molecule. Results at fixed equilibrium geometry and ~in parentheses for
l5` and l 5 632.8 nm! zero-point vibrational averages. Active space is
CAS-3 and basis set is h1. Atomic units.
a xx
Wavelength ~/nm!
`
2278.
1064.
694.3
632.8
590.0
514.5
488.8
7.80~7.83!
7.81
7.83
7.86
7.88~7.91!
7.89
7.91
7.93
a zz
14.57~14.70!
14.60
14.70
14.87
14.94~15.08!
15.00
15.14
15.21
Da
6.77~6.86!
6.79
6.87
7.01
7.06~7.18!
7.11
7.23
7.28
authors of Refs. 69,70 used the MCSCF response method to
compute equilibrium values, while Inoue et al.68 employed
CI wave functions and studied the behavior of a (2 v ; v ) at
varying internuclear distances.
Our zero-point vibrationally averaged mean static polarizability ~110.12 a.u.! compares favorably with the experimental value of 110.66 a.u.71 The same applies for the l
5632.8 nm value ~110.30!, to be compared with 110.78
a.u. given in Ref. 66. As far as the anisotropy is concerned,
the l5632.8 nm value of 17.18 should be compared with
the experimental value of 17.43 a.u. by Bridge and
Buckingham.66 Both Kling et al.3 and ~as we presume!
Lukins and Ritchie4 use, as said above, D a 517.36 a.u.,
again from Ref. 66.
Note that the limitations in our treatment of electron correlation or in the expansion of the basis set, which were seen
above to possibly influence the hypermagnetizability anisotropy, do not affect second order properties as the magnetizability and electric dipole polarizability anisotropies to any
appreciable extent, and the convergence with respect to both
electron correlation and basis set extension are satisfactorily
achieved with the CAS-3 reference wave function and the h1
basis set.
The diamagnetic contribution to the Cotton–Mouton
constant of molecular oxygen can now be computed through
Eq. ~6! @Eq. ~12!#. Table VIII shows our results for the static
regime and at the experimentally relevant frequency corresponding to l 5 632.8 nm. The product of the electric dipole
polarizability and magnetizability anisotropies, the quantity
which—together
with
the
hypermagnetizability
anisotropy—can be directly deduced from experiment,3,4 is
also given in Table VIII.
dia
The temperature dependent contribution to m C T→`
appears to completely overshadow the effect of the hypermagTABLE VIII. The diamagnetic contribution to the Cotton–Mouton constant,
dia
, at 293.15 K for the O2 molecule. Active space is CAS-3 and basis set
mC
5
22
is h1. Atomic units except m C dia
mol21 ).
T→` (m A
Wavelength
~/nm!
Dh
Dj
Da
DaDj
Q
~293.15 K!
dia
m C T→`
~293.15 K!
`
632.8
3.12
2.65
21.91
21.91
6.86
7.18
212.73
213.37
21881.8
21969.6
21.12310227
21.17310227
J. Chem. Phys., Vol. 106, No. 20, 22 May 1997
Jonsson et al.: Hypermagnetizability of molecular oxygen
8561
FIG. 1. The Cotton–Mouton constant of molecular oxygen as a function of temperature. Experimental data from Ref. 4. The error bars associated to our
para
curves for m C T→`
and m C tot
T→` correspond to standard deviations and are due to the error on the electron spin g-factors taken from Ref. 30.
netizability anisotropy, which gives less than 0.2% to the
overall values in the Table VIII. The contribution of the hypermagnetizability anisotropy is much smaller than predicted
from experiment, even when the huge error bars of Refs. 3,4
are taken into account. The results of the calculations are on
the other hand in line with those obtained for other molecular
systems as indicated earlier in this paper, and, if anything,
the contribution of D h to the CME seems for O2 even less
important than in other cases.
G. The Cotton–Mouton constant of O2
Figure 1 displays the temperature dependence of the india
para
dividual contributions ( m C T→`
and m C T→`
) and of the
Cotton–Mouton constant ( m C T→` ) of O2 at l 5 632.8 nm,
as extracted from the experiment.3,4 The experimental data
were taken from Figure 2 of Ref. 4. Due to the very close
relationships between the two available experimental results
~Refs. 3,4! we decided to include only the most recent data in
Figure 1. A comparison with the results of our ab initio
investigation is also indicated in Figure 1.
dia
Our curve for m C T→`
is plotted by employing Eq. ~12!
and the data in Table VIII. To compute the paramagnetic
contribution through Eq. ~13! we searched the literature for
reliable estimates of the components of the electron spin
g-factors of the O2 molecule. Minaev72,73 has published
semiempirical estimates of the electron spin g-factors of the
ground state of O2 . His MINDO/3 CI values are g z
52.0023 and g x 52.0049. The g-factors can in principle be
extracted from gas-phase electron spin resonance experiments, which are notoriously quite difficult to perform,74 and
to our knowledge such studies are rare ~but see Refs. 75–77,
29,78,79,30,80!. The most reliable result to date seems to be
the estimates of Evenson and Mizushima,30 e.g.,
g z 52.0020 60.0001 and g x 52.004460.0008. By applying
standard error theory and assuming perfect correlation between
the
two
components,
we
have
g 2z 2g 2x 520.009660.0036. This value was employed to get
para
in Figure 1. Also displayed is our esthe curve for m C T→`
timated Cotton–Mouton constant for the oxygen molecule
obtained employing electron spin g-factors of Evenson and
Mizushima. For a few temperatures we show with error bars
the standard deviation associated with the data.
The error associated with the anisotropy of the squared
electron spin g-factors extracted from experiment is huge,
amounting to about 37% of the anisotropy itself. This leaves
a large indeterminacy which transfers to our prediction of the
CME of molecular oxygen. The same argument applies in
our view to the reverse process, as applied by Kling et al.3
and adopted by Lukins and Ritchie4 to derive the diamagnetic contribution to the CME of O2 from their experimental
data, since Evenson and Mizushima electron spin g-factors
were also employed in that case. Note finally that the indeterminancy does not depend on the approximations made on
the magnetic dipole moment, i.e. on the neglect of fine structure terms mentioned at the end of Sec. II D. The effect
might not be negligible in principle when an anisotropy
J. Chem. Phys., Vol. 106, No. 20, 22 May 1997
8562
Jonsson et al.: Hypermagnetizability of molecular oxygen
(Dm 2 ) instead of squared magnetic moments are to be taken
into account.
IV. CONCLUSIONS
We have carried out an ab initio determination of the
hypermagnetizability anisotropy for the oxygen molecule, by
studying in detail the effect of electron correlation, basis set
extension, magnetic gauge origin dependence, frequency dispersion, zero-point vibrational averaging and pure vibrational contributions. Our estimated value at l5632.8 nm has
a different sign and is much smaller than the experimental
values,3,4 even when taking into account the experimental
error bars and the estimated residual inaccuracies due to possible incomplete treatment of electron correlation and convergence of the basis set. Those we could pessimistically
estimate to be about 40% of our ‘‘best’’ value D h 5 12.65
a.u. Also, our results, even if we cannot employ a magneticgauge-origin independent approach, show a very small dependence on the choice of the origin of the magnetic gauge.
Frequency dispersion appears to be quite small in the
wavelength range of experimental interest ~3.5%!. Zeropoint vibrational averaging corrects the equilibrium value of
the hypermagnetizability anisotropy by '6%. The effect of
the pure vibrational contribution to D h is noticeable
('60%! and is larger at finite laser frequencies than in the
static regime. All these effects are accurately included in our
estimation of the hypermagnetizability anisotropy of molecular oxygen.
The spin–orbit coupling between the ground triplet state
81
21
and the excited 1 S 1
g state is quite large (' 150 cm ), but
influenced the ground state hyperpolarizability only by ca.
1%.60 An accurate determination of the effect of spin–orbit
coupling on the hypermagnetizability of molecular oxygen
would require a consideration of singlet and triplet perturbations in cubic response theory ~as currently available up to
the quadratic response37!, and of a finite field technique. A
simpler approach, e.g., the use of first order perturbation as
that in Ref. 60, suggests, however, that spin–orbit coupling
should not influence the hypermagnetizability anisotropy
more than it does for the electric hyperpolarizability ~less
than 1%!, and thus that it will not be a good candidate to fill
the wide gap between theory and experiment.
We have to turn to other possible sources to try to explain the discrepancies that we obtain between theory and
experiment. If we accept the arguments put forward by Kling
et al.3 to affirm the validity of the high temperature limit
approximation for the expression of the Cotton–Mouton constant of O2 @Eq. ~1!# and assume that higher order terms in
the perturbative expansion leading to Eq. ~1! could be safely
neglected, we think that we are left with only one possible
source of uncertainty, i.e. the approximations used to relate
the paramagnetic contribution to the observable. It seems
quite probable at this stage that the approximation of Kling
para
et al.3 to m C T→`
might be off target. It relies on the approximations made by Tischer29 to obtain the wave functions
needed to describe accurately the fine structure of molecular
oxygen ground state27,31,50 and, perhaps more critically as
shown in the previous section, on electron spin g-factors30
which are not sufficiently accurate to be employed to deterpara
mine m C T→`
. Also it does not apparently follow the expected 1/T 2 relation @Eq. ~7!#. It thus seems risky to determine the magnetizability anisotropy, and—in view of its
negligible contribution to the effect—even more the hypermagnetizability anisotropy, by subtracting the paramagnetic
contribution from the experimental Cotton–Mouton constant. In turn, an ab initio prediction of the CME of molecular oxygen needs an accurate determination of the electron
spin g-factors, a task that seems at present far from trivial50
and which was certainly beyond the scope of the present
study.
ACKNOWLEDGMENTS
We thank David M. Bishop and Boris Minaev for valuable discussions.
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