Download Dynamic ligand-field theory for square planar transition metal

THEO
CHEM
ELSEVIER
Journal of Molecular
Dynamic
ligand-field
Structure (Theochem)
I ( 199X) 97- 107
theory for square planar transition
metal complexes
K. Wissing,
Imrrturfiir Theoreti.whP
43
Chrmw
J. Degen*
Hrinrich-Hrrnr-Unrvertitdt,
D-40225
LXisseldorf
German.~
Rcccivcd 19 August 1997: revised 6 November 1997; accepted 18 November 1997
Abstract
The electron-phonon
coupling of square planar transition metal complexes is analysed by a perturbative model based on the
electrostatic ligand-field theory. Excited-state potential energy surfaces are characterized by taking into account linear vibronic
coupling within the point group D4t,. Results do not depend on new adjustable parameters and can be generalized to a great
number of complexes. The applicability of the model is demonstrated by comparing the results of the model with those obtained
from the optical spectra of [CuC14] *- and [MX,] ‘- (where M = Pd, Pt and X = Cl, Br). In particular, the influence of E x (b ,a +
bz,) coupling, which is the subject of some controversy in the literature, is found to be of low importance. 0 1998 Elsevier
Science B.V.
Keywords: Vibronic
couplin g; Square planar complexes;
Band profiles; Transition metals
1. Introduction
Transition
metal complexes
have been of great
interest in the past because of their rich spectroscopy
in the optical region of the electromagnetic
spectrum,
which is mainly due to the open d-electron
shell.
Currently,
the search for new materials with special
optical and magnetical
properties
pushes the investigation in this area. This paper deals with the nuclear
geometry
of square planar complexes
in different
d-electron
states. Geometry changes following optical
excitation
are caused by changes in the metal-ligand
bonding
strength,
since valence
electrons
forming
the bonds are excited. The electrostatic
ligand-held
theory has provided an explanation
of band positions
in optical
spectra.
The width
and vibronic
fine
* Corresponding
author.
structure of these bands, if visible, have been treated
to date in most cases by symmetry
rules or by arguments concerning
orbital occupancy
changes in going
from one electronic
state to another.
A sufficient
qualitative
understanding
of the totally symmetric
displacements
in excited electronic
states is possible
by those arguments.
In the case of orbital-degenerate
electronic
states, a distortion
along a non-totally
symmetric
mode is demanded
by the Jahn-Teller
theorem. The strength of such couplings and the question of which mode dominates the coupling and determines the distortion often remain unclear. For square
planar complexes
there is still ambiguity
concerning
the question of the importance
of couplings of unsymmetrical modes with d-electron
states and the relative
importance
compared with the coupling for the totally
symmetric
mode.
A model for the computation
of spectroscopic
band
0166-1280/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved.
PII SO166-1280(97)00433-8
98
K. Wissing,
.I. Degn/Journal
c$Molrculcrr
profiles and nuclear distortions is developed here,
which is as easy in application as the electrostatic
ligand-field model (LFM) for the computation
of
band positions. It is an extension of the dynamic
ligand-field model for octahedral complexes [ 1,2] to
the lower, square planar geometry. There is no need
to introduce new free adjustable parameters
the
necessary information comes from vibrational spectra
(frequencies), the metal-ligand
distance and static
ligand-held
parameters. The mixing effect of the
nd,? orbital with the (n + 1)s orbital on the metal (sd
mixing), both having a,, symmetry in square planar
complexes, is taken into account by its influence on
electron-phonon
coupling.
The basic magnitudes in vibronic coupling theories
[3] are the vibronic coupling constants, which quantify the force tending to distort the complex from a
high-symmetry
reference geometry along a normal
coordinate, depending on the electronic state of the
complex. Many attempts have been undertaken in
the past to determine these constants by very different
approaches. Determination of first-order Jahn-Teller
coupling constants is enabled by knowing the amount
of splitting of the electronic energy levels caused by
symmetry
lowering owing to nuclear distortions
along non-totally
symmetric modes. Therefore, by
means of electronic structure calculations of arbitrary
type varying the nuclear geometry, these constants
are derived and, from the experimental
viewpoint,
vibrational progressions in optical spectra are usually
correlated by vibronic coupling constants to nuclear
distortions. The theoretical foundation of vibronic
coupling theory is found in [3] and references therein.
A parametrization
scheme of vibronic coupling constants in terms of the angular overlap model (AOM)
established by Bacci [4,5], is related to the present
approach as discussed below.
The idea of the model presented is to calculate
vibronic coupling constants starting from the electrostatic LFM by computing the first-order term in the
Herzberg-Teller
expansion [3] and treat this operator
(linear vibronic coupling operator) as a perturbation
on the d-electron states:
3N-6
H’ =
iz
(~H”/~Qi),Qi
Ho is the electronic
the nuclear reference
Hamiltonian
of the system at
geometry, Qo, consisting of the
Structure
Fig.
(Theochem)
I.
Repulsive
431 (1998)
interaction
97-107
of an electron
ligands tending to distort the complex
nuclear coordinate
by a dynamic
(e, x /3, coupling
LFM calculation
in an ep orbital
in the direction
on the
of the h ,e ((3 ,)
case), which can be quantified
of the 0; coupling
constant.
kinetic energy (T,) and the potential energy contributions of electrons (e), ligands (L) and the metal ion
(M):
~“=T,+v,,+v,,+v,,(+Hso)
(2)
Hso is the spin-orbit
coupling operator. Apart from
the electrostatic ligand-field potential, VeL, the other
terms in Eq. (2) depend only weakly on nuclear geometry changes. The vibronic coupling operator is
therefore computed as the derivative of the electrostatic ligand-field potential with respect to the nuclear
coordinates at the square planar reference geometry.
Fig. 1 provides a visual example of the repulsive
force imposed by a d electron on the ligands in an
eg x /3, coupling case, taken into consideration
by
the dynamic ligand-field model.
Although electrostatic ligand-held theory approximates the first ligand sphere only as point charges
and limits the basis to pure d-electron functions on
the central atom, semi-empirical
treatment of the
radial parts of the integrals has been applied to a
wide range of systems with great success [6]. In the
following, the orbital and many-electron
d* and d’
coupling cases as well as the influence of sd mixing
on vibronic coupling constants are treated. Then, a
comparison of the vibronic coupling predicted by
the model and that obtained from well-resolved
vibronic spectra shows that the semi-empirical treatment of the (same number of) parameters validates
the model assumptions also in this extension of the
LF model direction or to electron-phonon
coupling.
2. Orbital coupling case
In Fig. 2 the splitting of the d orbitals in a square
planar D4,, complex, including the influence of sd
K. Wissing, J. DegdJoumal
oj’kfolecular
mixing, is shown. The relative energy of the dLz orbital
depends strongly on the extent of sd mixing. We start
with the pure d orbitals, coupling with the modes of
the square planar D4,, complex. Only vibrations with
even symmetry are able to couple to states originating
from a d” configuration. The symmetrized Cartesian
displacement coordinates of the vibrations in question
are [3]:
b,,(P,)
Qp, =;iW2-X?+Yd
x=0
Structure
(Themhem)
99
431 (199X) 97-107
8 and 4 are electronic coordinates. 2 is the effective
charge on the ligand. Since the ligand-field potential
depends on polar coordinates, coordinate transformations are necessary [l] (aV,L/aQi)o’(aV,L/aX,)o,
(aveL/aY,)
+
O~(aveL/azjh
(aveLlaRjkJ3
(aveLia@j)O,
(IFIV,L/C~@,)O
(i = 0$,,/32; j = 1...4). The coupling
operators are calculated at the high-symmetry
(Da,,)
nuclear configuration
up to the fourth multipole
(X 5 4) of the electrostatic ligand-field expansion.
Higher multipoles are of no interest here, because
they give no contributions to matrix elements over d
functions.
The derivatives of the ligand-field potential with
respect to the coordinates in Eq. (3) are:
A=2
+6
-
--
=
(3
6
+...
5
__
b2$32)
_
z
Qp2 = ;iyl
+x2
+ $G$r4i(Y~
- y3 -x4)
(3)
The numbering
of the ligands
is 1,2 on the
positive x,y axes and 3,4 on the negative x,y axes,
respectively.
The orbital coupling operators are calculated as
the derivative of the electrostatic ligand-field potential
for localized d electrons (given as an expansion in
spherical harmonics) [6]
A=0 2x+ 1
r=X
(4)
with respect to the symmetrized Cartesian displacements in Eq. (3) at the square planar nuclear
geometry. R, 0 and + are nuclear coordinates and r,
+ ...
+ Y4m2)
The indices of V are dropped. For reasons of symmetry, the following coupling cases are possible in D4,,
symmetry:
Ulg
x (2
h, XQ!
eg
x
b,, x Q!
(a,,+&,)x01
(cu+P,
(a,,+b2p)
+P2)
XL%
The perturbation matrix elements with the coupling
operator are related by the Wigner-Eckart
theorem
[3] for a given coupling case. Only one of the matrix
elements has to be computed in each coupling case
to give the reduced matrix element, which is proportional to the vibronic coupling constant. When the
ligand-field parameters ~0, r~ and Dq are introduced
[61
Dq= A$(r’)
the orbital
coupling
constants
(6)
for a square planar
K. Wissing, J. DegdJmrnal
100
of Moleculur Structure (Theochem) 431 (1998) 97-107
complex are:
(7)
u”,‘, for example,
equals
the matrix
element
(d;l ((tlV/6Q,),jd,2),
the upper index (a ,) denoting
the symmetry of the electronic function d,z and the
lower one denoting the type of coupling mode,
which is the totally symmetric stretching vibration
in this case. For vibronic coupling with the doublydegenerate electronic e state and for the higher-order
coupling cases, the corresponding coupling matrices
are also given. Since the LF potential, VeL, contains
only the repulsive interaction between ligands and d
electrons and no metal-ligand
attraction, the groundstate (Y coupling constant is negative and not zero
as demanded. To compute excited-state 01 coupling
constants, the computed value for the ground-state
coupling constant has to be subtracted from the
excited-state values. This has the same influence as
the introduction of an attractive metal-ligand
potential in VeL. The terms with co, representing
the
spherical symmetric part of the ligand-field potential,
vanish as a consequence. Stabilization energies and
nuclear distortions due to linear (Y,/3, and fi2 coupling
cases are (a! are the corrected constants):
AE’ = - (a3*/2k,
with F=u,~,
and
b,,, bZg, e
AQ, = -aL/k,
(8)
ki being the force constant of the vibration i. In the
linear e x (/31 + 01) coupling case AQ, can have both
signs and there are two kinds of possible minima,
either two e x /3, or two e x /Z2 minima. According
to 6pik and Pryce [7], the deeper of the two kinds of
minima are absolute minima whereas the other two
are saddle points. The force constants can easily be
calculated from the vibrational frequencies:
k; = 47r2c2piif
X (~58
991
X
IO-‘.~i
[g/mol].#
[cmm2] kg/s2)
(9)
The reduced masses, pI, are the ligand masses
considering the choice of coordinates in Eq. (3).
For calculation of the coupling constants in Eq. (7),
the formal equivalence between the AOM and the
dynamic LFM is exploited by relating the parameters
Dq and 9 to the well-known AOM parameters e, and
e, F31:
lODq=3e,-4e,,
3 3-4e,/e,
‘= 5
1 +e,/e,
(10)
By this means, treatment of n (which is unknown in
magnitude) as an adjustable parameter is avoided.
Vibronic coupling constants in terms of AOM parameters (e,, e,, Ge$hR, Ge,JAR) have been derived by
Bacci [4,5]. For bending vibrations these constants
are equivalent to the constants in Eq. (7) obeying
Eq. (10) and can be regarded as the counterpart
of AOM coupling constants in terms of the LFM.
Coupling constants for stretching vibrations require
two more parameters in the AOM (Ge$6R, &e@R)
and differ from the constants given here because
of diverging model assumptions concerning the distance dependence of the metal-ligand
interaction. In
K. Wining, J. Degen/Journal
of Molecular Structure (Theochem) 431 (1998) 97-107
‘- n+1s
h
Ig:
R’
d+2
D4h
; b,, dXz-+
sd-Mixing
Fig. 2. Term scheme for a d’ system, showing the influence of sd
mixing.
fact, the indirect semi-empirical
treatment of 7 as
suggested by Eq. (10) is preferable, since it is well
known that the calculation of ligand-field parameters
such as Dq from first principles gives results that are
an order of magnitude too small. It has been shown,
however, that the dependence of Dq on metal-ligand
distance predicted by ligand-field theory to be
proportional
to Rm5 - compares fairly well with
experimental
observations
for various transition
metal complexes and corresponds to other types of
computational
method [9,10], which is a condition
for successful application of the dynamic LFM for
computation
of the vibronic coupling constants of
stretching modes.
3. Many-electron
configurations
coupling cases for d8 and d9
The coupling constants for multiplets resulting
from many electrons in the open d shell are a linear
combination of orbital coupling constants, since the
vibronic coupling operator is a one-electron operator.
Table 1
Many-electron
coupling
constants
for d8 strong-field
101
Most known square planar transition metal complexes have a d8 or d’ electron configuration. Without
regarding sd mixing, a d9 complex has the opposite
term scheme from that depicted in Fig. 2, owing to
the electron hole formalism. Likewise, the orbital
coupling constants derived above are directly applicable to d” complexes when all signs of the constants
are changed. It should be noted that, in the linear
coupling case, the signs of 0, and p2 coupling constants are of minor importance, because distortions
along both directions of one coordinate are equivalent.
This is different to the situation for an octahedral
complex coupling with an E vibration, for example,
where the sign decides whether the complex is tetragonally elongated or compressed.
Regarding vibronic coupling with the totally symmetric mode, the CYcoupling constant of the ground
state (GS) is subtracted from excited-state (ES) 01
coupling constants as in the orbital coupling case.
The constants depend only on the occupation of the
d orbitals:
A:”= ,;
(d:"a;),
AzS = ;g (&z;)
(11)
The sum runs over the 10 d spin orbitals and d, is the
occupation number of the spin orbital i. Eq. (11) is
also valid for configuration
interaction
(CI) base
functions in the d( + s) basis.
The construction of linear vibronic coupling constants for many-electron
strong-field states with Pi
and p2 modes is limited to electronic E states, because
non-degenerate
states cannot couple to non-totally
symmetric modes in the linear case. The E x PI and
E x /3? coupling constants for the six occurring E
states in a d* configuration are computed by Griffith’s
irreducible tensor method [ 1 l] and listed in Table 1.
Non-diagonal
matrix elements between different E
strong-field states are all zero, as can be seen from
Griffith’s formulas. Regarding CI in the d( + s) basis,
owing to non-diagonal
elements of the electron
E states and non-totally
symmetric
modes reduced to orbital coupling constants
102
K. Wissing, J. DegedJoumal
of MolecularStructure
repulsion and the spin-orbit
coupling operator in
Eq. (1) (static problem), a coupling constant for a CI
basis function, ‘I!“= cicl\kE,i (i = 1...6), is:
(Theochem) 431 (1998) 97-107
orbital is a 2 x 2 problem for a square planar complex
[ 12,131. The perturbation matrix of the static problem
is:
(14)
A;;‘,, 42 is the coupling constant for matrix elements
over strong-field states qEE.,,which can be simplified
to orbital coupling constants with Table 1.
In d8 complexes non-diagonal coupling constants
between different strong-field states only occur for
non-degenerate states. The effect should be significant
for large coupling constants between states, which
are close in energy (if kAE/4a2 < 1, the lower potential has two minima). The second-order coupling case
* “b’zgb’ig)+“B,,(al,e~b&blJ]
X f12 involving
]3&g(argee
the two lowest-lying
non-degenerate
triplet states
may be of interest:
‘A,,‘B,
Al%-
’
‘A,,jB,
A&-
_ a,.hz
-a@:
(13)
The effect of sd mixing has been found to be important in the interpretation of the electronic energies in
square planar complexes. It is therefore necessary to
discuss its influence on the linear vibronic coupling.
sd mixing causes an enlargement of the dz2 orbital
in the z direction and a decrease in the x and y directions. Other orbitals are not influenced for symmetry
reasons. Excited-state energies relative to the ground
state are changed when the occupation of the di2
orbital is changed. Considering
linear coupling
cases, electrons in a d,z orbital are not able to contribute to couplings
with non-totally
symmetric
modes. The influence of sd mixing is then limited to
the a:’ coupling constant and causes it to be less
negative.
The interaction
(15)
Since there are four ligands lying in the .xy plane, the
HSd’ matrix element is:
4. Mixing of nd and (n + 1)s functions at the central
atom (sd mixing)
4.1. Quantitative
Hsd’ equals (sl V,Jdzz), where Vex is the static ligandfield operator given in Eq. (4), and AEYdis the energy
difference between the s and d.1 orbitals. Most commonly in AOM treatments the parameter (J,d is introduced, which describes sd mixing between the s and
dz: orbitals caused by the potential VLL of one ligand
lying in the z direction:
F,,dcZ(8, 4) is a geOIIEtriC
f&Or
[14]
and HJd’ iS
negative as can be seen from Eq. (19). Assuming
that AE,Yd >> ksd’ 1, the perturbation
energy for
can
be
approximated
by
the
dzz
orbital
The eigenEsd’ = -40,~~ [ESd(d$) = E(d,z) -4axd].
vector for the perturbed dz2 orbital depends only
on the ratio p = AEld/I?rsdt and can be given analytically ( - m < p I 0):
d;;”= cl d,z + c2s
and
(17)
treatment
between the nd,z and the (n + 1)s
This function, used as basis function for the perturbation treatment with the vibronic coupling operator
K. Wining,
(aV/aQ,),,
J. DegenlJournal
of Molecular
Structure
(Theochem)
the treatment of 11’as a free adjustable parameter. The
corrected (Ycoupling constant of the dzz orbital with s
admixtures is computed from Eqs. (17), (18) and
(20). Common AOM parameters such as e,, e, and
(T,dare sufficient to account also for the influence of sd
mixing on potential energy shifts.
yields:
5. Comparison
To derive an expression for p in Eq. (17) in terms of
common ligand-field parameters, we can write, with
Eq. (15): p = Hsd’lusd. The integral Hsd’ is computed
by usual ligand-field theory to be
(19)
so that p is:
12 Dq
(20)
p=-JJijTg
If Eqs. (16) and (19) are used, an expression
can be given as:
for ?I’
(21)
which depends only weakly on the unknown parameter AEsd. If we assume typical values for AE [ 151
( = 80000 cm-‘) we are able to include the influence
of sd mixing in dynamic ligand-field theory, avoiding
of computed
“Calculated
results and parameters
obtained from the xy-polarized
Ak$
absorption
AQZ” (pm)
AQyP[16]
17
16
15.9
- 1130
- 1070
11,13”
21.2
- 450, - 690’
with the inclusion of sd mixing (7’ = 0.30).
with vibronic
spectra
We shall apply the theory developed so far to the
square planar tetrachloride of Cu*+ [ 15,161 and to
the tetrachlorides and tetrabromides of Pt2+ and Pd*’
[ 17-261, comparing it with well-resolved
vibronic
spectra reported in the literature. The ground-state
vibrational frequencies [ 16,271 of the chromophores
are taken for computation of the force constants with
Eq. (9) which are assumed to be valid also for excited
states.
The (metH)2CuC14 (metH = bis(methadonium))
complex has a d9 configuration giving rise to three
ligand-field transitions, which are clearly seen in the
xy-polarized spectrum [16]. Adaptation of the AOM
parameters to the positions of these bands yields e, =
5270 cm-‘, e, = 920 cm-’ and aBd= 1500 cm-‘. With
the vibrational frequencies [ 161 (V, = 275 cm-‘, VP, =
195 cm-‘, 3p, = 181 cm-‘) and R = 227 pm (according
to [ 161 there is a very slight deviation from the square
resulting
in small differences
planar geometry,
between R values; 227 pm is an average value), the
results given in Table 2 are obtained and compared
with experimental values.
The Pd*’ and Pt’+ complexes have a d8 configuration and the term splittings under the influence of
different interactions are depicted in Fig. 3. Careful
AOM analysis of these complexes has been performed
in [28], and the values for e, and e, are overtaken
with:
Table 2
Comparison
103
431 (1998) 97-107
(cm-‘)
spectrum of the (metH)zCuClj
complex
sc”1’
a
(% ( SP,)
S:p[16]
4.1
3.9
(0.005, 0.07)
1.7, 2.5”
-4
-4
-6
K. Wissing. .I. Degen/Journul
104
alg bl,
, 1,” .’
\
I
of Molecular Structure (Themhem)
Huang-Rhys
factors (S = AE/~v,,~) are compared
with experimental values; this is a more sensible test
for the model than the comparison of nuclear distortions, because the first depend on the square of and
the latter depend linearly on the coupling constant.
The determination
of the Huang-Rhys
factors
from the electronic spectra is confined to transitions
showing well-resolved
vibrational
fine structure.
Hence transitions to the ‘E, and ‘B,, states are not
evaluated. The Huang-Rhys
factor is identified with
the number of the most intensive progression sideband. If that number or the energy of the O-O transition is not given in the reference, we have estimated it.
In Table 4 the observed values are given and it is
indicated whether the number is obtained from an
emission or an absorption spectrum. If the reported
spectra are polarized, we only analysed the .ry-polarized spectrum because of its usually higher resolution
and owing to the fact that the transition to the ‘Azg
state is only visible in xy polarization. The spectra in
[20,22,23] are microphotometer tracings.
‘Es
‘Al,
R3
1. D4,, , sd_mixing
2. Electr.-electr.
interaction
Fig. 3. Section of the term scheme for a square planar dX complex,
visualizing the different states of approximation in the perturbation
treatment. The configurations used at approximation
level I are
holes in the d shell.
(Table 3) for our calculation and assumed to be valid
for all compounds containing the chromophore in
question. For example, we cannot distinguish between
K2PtCI, and the doped compound, Cs,ZrC& : PtCl$- ,
in our calculation.
Cl is neglected for the results
in Table 4. No higher-order coupling between different strong-field states is regarded. At this level of
approximation’,
results are equal for singlet and
triplet strong-field
states with the same electron
configuration.
Parameters from the static problem necessary for
the calculation and values derived from the equations
above are listed in Table 3. In Table 4 calculated
6. Discussion
The expressions for the vibronic coupling constants permit some general remarks to be made.
From Eq. (5) it is seen that the relative magnitudes
of the coupling constants depend only on the parameter 7, which depends on the ratio eJe, through
Eq. (10). Reasonable values of this ratio for transition
metal complexes give 17values between 0.7 and 2.5.
In the strong-field approximation
the lowest electronic transitions
for d8 and d9 complexes
are
one-electron
jumps to the b,, orbital (big - b,,
’ The strong field states of approximation level 2 in Fig. 3 serve as
basis functions for the perturbation treatment with the vibronic
coupling operator in Eq. (4).
Table 3
Literature parameters
for dX complexes necessary for the calculations
431 (1998) 97-107
(upper part) and values derived from the formulae given in the text (lower
part)
e,, r, WI (cm-‘)
fi,, VP,, Cal~271 (cm-‘)
Dq (cm-‘). R [27] (A)
4. v’
k,, k,#. kO>(kg/s’)
[PtCl‘J ?-
[PtBr4]‘-
[PdClq12-
[PdBr,] ‘-
12 420, 2800
329, 304, 196
2606, 2.308
1.03, 0.44
226, 193, 80
10920, 2200
205, 188, I28
2396, 2.445
1.10, 0.44
198, 167, 77
IO 150, 2000
307, 274, 198
2245, 2.313
I.1 I, 0.42
197, 157, 82
9510, 1780
190, 171, 127
2141. 2.444
1.14. 0.42
170, 138. 76
-56, -39, -46
-57, 3.6, -3.6
-49, -32, -39
-48, 2.1, -3.3
-48, -32, -38
48, 1.9, -3.3
-44, -28. -34
-43, 1.3, -3.0
K. Wining,
Table 4
Comparison
of computed
J. Degem’Journal
and experimental
Calc.
Exp.
S,(‘n,,,**
.S,(‘Blo)
II
5, 7*
S,( ‘A
II
S,(E&**
SD,. S/3?
II
0.06, 0.2
Structure
(Theochem)
Calc.
6 [25,29] L.p IS
4 [20] “,P
6. 9*
4 [20.22] C.d
8 [25] rG
5 [2l] “.h.p
10
4, 6*
7 [20] a.h.p
8 [2l] r.h.p
IO
I.5
I05
97-107
[PdBr4] ‘-
[PdCIJ’Exp.
Calc.
431 (1998)
factors estimated from literature spectra of dX complexes
[PtBr,]‘-
[PtCI,]‘-
:,)**
Huang-Rhys
of Molecular
I5
0.03, 0.3
Exp.
6 [23] “.’
6 [26] ‘W
10
0.02, 0.2
Calc.
Exp.
I5
6. 9*
,u ,241 “h.P***
I5
8 [24] a,~+
I4
0.02, 0.2
L- Absorption spectrum. ’ - emission spectrum. et - excitation spectrum.
“Estimated value (without band analysis).
Calculated from the Stokes shift (if absorption and emission spectra are available): S = (A.Es, - 2~,,)/2v, (Ven is the frequency of the enabling
mode).
“Pure potassium salt.
“Doped compound - Cs2ZrCI,:MC$
(M = Pt or Pd).
‘Calculated with the inclusion of sd mixing (AESd 1s set at 80000 cm-’ [IS], and erd = l/4e, [28]).
“*At this level of approximation (level 2 in Fig. 3) the same numbers are computed for states differing only in their multiplicity.
‘**According to the calculations in [2X]. the spacing of the two spin orbit split states, Pj and PY of the ‘B,, state, is about 430 cm-‘. The
observed long progression could therefore be due to two electronic transitions.
(1ODq jump), e, - h tp and a Ig - b Ig, ordered according to increasing
energy). From Eq. (12) the Q!
czupling constants for these orbital transitions are
Ii
&Y -a, r (I? = bsg, eE and alp) and Table 1 shows
that the /3, and pz constants for strong-field E states
of d8 ions correspond also to the orbital coupling
constants (the sign is of minor importance in a linear
coupling, because the direction of the distortions is
equivalent). That is why a plot of the d” coupling
constants against 7, setting DqlR = 1 (Fig. 4), reveals
information
concerning
the relative
strength of
different coupling cases in ds and d’ complexes.
The following results are obtained for complexes
with d8 and d” configurations.
Result 1: Concerning
electronic E states, the
absolute /3, and /32 coupling constants are at
most l/S and l/8 of the absolute cy constant for
reasonable 11 values. Assuming Ila/t)a, = 1 and
WPz = 3/2 for the vibrational frequencies, one
can compute S, 2 25S,, and S, > 18S0, for the
Huang-Rhys
factors with Eq. (8). Consequently,
in square planar complexes,
linear vibronic
coupling with non-totally
symmetric modes is
supposed to be of minor importance with regard
to its contribution to the Stokes shift.
Result 2: The Q! coupling constants of states
resulting from ulg - b,, transitions are smaller
than the ones from the other orbital transitions,
although the inclusion of sd mixing reduces the
difference. This result is not in perfect agreement
with the spectrum of CU” (see below).
Result 3: Higher-order couplings with the PI
d9
A; for
DqlR =
dX
1
Fig. 4. Plot of d’ coupling constants (d’ coupling constants with
negative sign) against the parameter 1, setting DqlR = I. The d8
coupling constants indicated correspond to the dX strong-field states
in Fig. 3 (multiplicity is not important). The QI coupling constant
of the ground state is subtracted from the other 01constants and set
to zero (see text).
106
K.
Wining,
J. DegerdJournal
of Molecular
mode should have a low influence because of
the small coupling constant and higher-order /3,
couplings are also not favourable, because in d8
and d’ complexes there are no low-lying electronic states close in energy that are able to
couple to this mode.
Result 4: The (Y coupling constant of the 1ODq
jump does not depend on 17.
Regarding the results obtained from optical spectra
given in Table 2 for the Cu’+ complex, a considerable
correspondence between computed and experimental
nuclear displacements
and Huang-Rhys
factors for
the 2B2B and ‘Eg states is found. The ‘A lg band is
calculated to be too narrow even if the effect of sd
mixing, which tends to broaden this band, is considered. The width of the 2E, band should, in principle,
arise from a coupling corresponding to Result 1, the
unsymmetrical
modes could be responsible
for
the missing resolution of vibrational fine structure.
The spectra of Pd2+ and Pt2+ complexes reveal good
agreement between theory and experiment (Table 4).
First, it is correctly predicted that vibrational progressions with the totally symmetric mode, observable in transitions involving the ‘Big state (a Ig - b Ig
jump), are shorter than progressions involving ‘Azg
and ‘Azg states (bzg - b,, jump). In contrast to the
Cu” complex, Result 2 agrees with the progression
length in the spectra. From Table 4 the Huang-Rhys
factors for E, states should be similar to those for Alg
states. In fact, similar half-widths are observed for
absorption bands to these states. Upon changing
from the chloride
to the bromide
ligand,
it
is correctly predicted that the progression becomes
longer. Going from Pt2+ to Pd2+ no significant effect
on the band width is observed either in the calculation
or in the spectra. Including the sd mixing effect
yields an increase in the progression length calculated
for transitions between the ground state and the jBlg
state (Result 2). The calculated absolute values in
Table 4 are a bit too large, but of the right order of
magnitude. Configuration interactions due to electron
repulsion and the spin-orbit coupling operator result
only in small changes of the computed results.
Huang-Rhys
factors in most cases differ by less
than unity for the spin-orbit
split states compared
with the strong-field parent states. This is reasonable,
because of the comparatively strong influence of the
Structure
(Theochem)
431
(1998)
97-107
ligand field in relation to octahedral and tetrahedral
complexes. Corresponding to Result 1, the broadness
of the E, bands is supposed to be due to (Ycouplings.
In [29,30] unusual spectroscopic
features have
been revealed for K?PtCl, spectra. First, an energy
gap between the absorption and emission spectrum
of about 1800 cm-’ with no spectral intensity is
found and, second, the vibrational frequency in the
emission spectra lies between the ground-state frequencies of the CYand PI stretching modes. By using
time-dependent
theory of electronic
spectroscopy
these features
are explained
by proposing
an
excited-state potential in the Qp, direction depending
on the three adjustable parameters k,tfjw,rr), a and A:
v(Qp,) = &VP;,
+A exp( -a’Qi,)
(22)
Optimization of Eq. (22) yields a distortion along the
/3, stretching mode of the same order of magnitude
as along the (Ymode (AQ, = 12 pm, AQp, = 12 pm in
[29] and AQ, = 12 pm, AQa, = 16 pm in [30]). The
progression
frequency
in the emission
spectrum
can then be explained by the missing mode effect
(MIME). The excited state has not been specified.
Although our simple model is not able to explain
the unusual spectroscopic features, such a strong 6,
distortion seems unlikely on the basic assumption of
our model: i.e., that the excited-state distortion is
caused by a change in the electron distribution in
the d shell of the central ion. We use a limited basis
and consider only the first ligand sphere, which however is necessary in semi-empirical ligand-field theory
in order not to increase the number of parameters
too much. In fact, the present approach allows for
the calculation of nuclear distortions, band shapes
and stabilization energies on the basis of common
parameters derived from static ligand-field or AOM
analysis of optical spectra with no additional free
adjustable parameters to be fitted to experiment. On
the other hand, that is why quantitative correspondence cannot be expected.
7. Conclusion
A perturbative treatment of the electron-phonon
coupling of square planar transition metal complexes
allows for characterization
of the potential energy
K. Wissing, J. DegedJournul
of Molecular
minima of electronic states within the open d shell.
For square planar dx and d9 complexes, vibronic
coupling of degenerate electronic states to /31 and 01
modes is expected to be negligible compared with
coupling to the totally symmetric vibration. Results
obtained from analysis of optical spectra exhibiting
vibronic structure are in good agreement with the
predictions of the model. Thus the method may be
used to help interpret optical spectra with less wellresolved structure. It is not too elaborate to extend
the model to systems of other symmetry. Tetrahedral
symmetry, which makes the treatment of three JahnTeller-active degenerate modes necessary, might be
of interest.
Acknowledgements
Financial support from the Deutsche Forschungsgemeinschaft
and from the Fond der Chemischen
Industrie is gratefully acknowledged.
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