Vibronic structure induced in spin-forbidden transitions in emission

Oct 1, 1992 - Jenny V. Lockard, Guadalupe Valverde, Daniel Neuhauser, and Jeffrey I. Zink , Yun Luo, Michael N. Weaver, and Stephen F. Nelsen...
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J. Phys. Chem. 1992, 96, 8157-8165

8757

Vibronic Structure Induced in Spin-Forbidden Transitions in Emission and Absorption Spectra by Excited-State Coupling David Wexler, Jeffrey I. Zink,*

Department of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles, California 90024 and Christian Reber*

Department of Chemistry, University of Montreal, Montreal, Quebec, Canada H3C 3J7 (Received: April 9, 1992; In Final Form: June 25, 1992)

Spectroscopic effects of spin-orbit coupling of excited-state potential surfaces are calculated by using the numerical integration of the time-dependent Schrodinger equation and the time-dependent theory of electronic spectroscopy. Intensity borrowing by a spin-forbidden transition from a nearby spin-allowed transition is calculated in terms of amplitude transfer of the wave packet between states. The main emphasis of the calculations is to analyze the vibronic structure in emission and absorption spectra arising from coupled surfaces. The coupling causes dramatic changes in both the relative intensities of the vibronic bands and the spacings between members of a progression. These changes are quantitatively calculated, and the theory is applied to the spectra of transition-metal complexes. The intensity and spacing between vibronic peaks in the absorption spectrum of K2Ni02are calculated and analyzed. A striking example of relative intensities in vibronic peaks induced by spin-orbit coupling is found in the emission spectra of d2 and d3 metal ions in octahedral environments where the lowest energy spin-forbiddentransitions arise from a change in the spin state with no change in the orbital component. Short progressions in totally symmetric modes are frequently observed even though no changes in the orbital populations, bond properties, or force constants are expected. The vibronic structure in spectra of Ti2+,V3+,Cr3+,and Mn4+ ions in octahedral halide lattices is analyzed.

1. Introduction

The electronic spectra of transition-metal compounds form a rich area for investigatingthe effects of potential surface coupling because one of the important sources of state coupling, spin-orbit coupling, is large and because frequently many excited electronic states with different displacements and force constants lie relatively close together in energy.’ One of the most important spectroscopic consequences of spin-orbit coupling is “intensity borrowing” in which a formally spin-forbidden transition “borrows” or “steals” intensity from a spin-allowed transition. (From the perspective of the forbidden transition, the intensity is borrowed, whereas from the perspective of the allowed transition, the intensity is stolen.) The appearance of spin-forbidden transitions in the spectra of metal complexes is frequently explained in this way.24 Another important consequence of spin-orbit coupling is unexpected vibronic structure. Progressions in a vibrational normal mode may be induced in forbidden transitions, or the relative intensities of the members of a progression may take on unusual patterns. These effects may reveal themselves in the band envelopes in unresolved spectra or most interestingly in the vibronic bands themselves in resolved spectra. Discussions of intensity borrowing by spin-orbit coupling of two states of different spin multiplicity usually focus on the electronic wave functions. The emphases are on how much of one electronic wave function is mixed into the second and the resultant effect of the mixing on the oscillator strength of the formerly forbidden transition.” In the usual fmt-order perturbation theory treatment, the amount of the allowed (qa) character in the forbidden (qf) state is equal to2

If the oscillator strength for a transition from the ground state to the allowed state is fa, the oscillator strength for the forbidden transition will be

Et ahd E, are the energies of the forbidden and allowed electronic transitions, respectively. The spin-orbit perturbation scrambles 0022-3654/92/2096-8157$03.00/0

the allowed and forbidden excited states and enables the spinforbidden transition to borrow intensity from the spin-allowed transition. Superficially, it might appear from the perturbation theory expression that the Franck-Condon factors and hence the vibronic structure would not be affected, an expectation far from the actual result. The quantum mechanical calculation is not trivial because the coupling of potential surfaces along normal coordinates results in a situation where the Born-Oppenheimer separability of nuclear and electronic wave functions cannot be made. The best-known examples of vibronic structure that is strongly affected by coupling are in the absorption and emission spectra of intraconfigurational transitions of metal complexes. The prototypical example is the lowest energy doublet state of d3 metal ions in octahedral The 4A2, ground state and the 2E, excited state both arise from the t22 electron configuration but differ by spin. Because the oneelectron transition does not involve a change of orbital population, any bond length changes or force constant changes between the ground and excited electronic states are expected to be very small or zero. Thus, most of the vibronic intensity is expected to be found in one line with zero quanta of vibrational excitation (excluding any enabling modes); i.e., no progressions in any of the normal modes would be expected. Frequently, however, short progressions in totally symmetric modes are observed.6 What causes these progressions? A similar question arises for octahedral d2 metal ions where the 3Tlgground state and the ITZgexcited state differ only by spin.* Again, short progressions may be observed. In this paper, we calculate how an electronic transition to a state with a spin different from that of the initial state acquires intensity by spin-orbit coupling with a third state. We emphasize how the vibronic structure in the “forbidden” transition is affected by the coupling. The emission and absorption spectra are calculated exactly in one vibrational dimension by using the recently developed split-operator techniques of Feit et ala9These methods make it possible to calculate the wave-packet dynamics for any potential surfaces. The physical picture as well as the quantitative details is discussed. The theory is applied to the resolved vibronic structure in the absorption spectrum of K2Ni02and the emission spectra of Ti2+, V3+,Cr3+, and Mn4+ ions in octahedral halide lattices. 0 1992 American Chemical Society

8758 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 2. Theory

The time-dependent theory of electronic spectroscopy has been discussed in detail previously, and only a brief outline of the relevant aspects will be given here. The absorption spectrum is given byloqll I ( w ) = C o ~ ~ m e x p ( i o t ) I ( 4 1 4 (exp( t))

-r2t2+ 7t)) dt

(3) where I(@)is the absorption intensity at frequency o,Eothe energy of the electronic origin transition, and I' a phenomenological Gaussian damping factor. The emission spectrum is calculated by using a similar equation I(u) = Cw31+mexp(iwt) -OD

(4)

The most important quantity is (414(t) ), the autocorrelation function of the initial wave packet 4 prepared on the final electronic state potential surface after the spectroscopic transition. The spectra are governed by the dynamics of a wave packet on potential surfaces. The initial wave packet 4i= 4i(t=O) = piX(Q) with i = 1,2 is the vibrational eigenfunction x(Q) of the imtial electronic state multiplied by the transition dipole moment p for the transition to the final electronic state. The wave packet + ( t ) evolves on the latter surface with time. For absorpion transitions to two coupled excited states, two wave packets, and &, moving on the two coupled potential surfaces are needed.I2-l9 The total overlap (4l$(t))is

(4l4(t)) = (4ll4dt)) + (42142(0) (5) The propagating wave function t#,(t) is given by the thedependent Schrainger equation. For one state i in dimensionless form, it is9 i

84 1 = --V2& at 2~

+ V,(Q)& = Hi+,

where Hi denotes the Hamiltonian, V,(Q) is the potential energy as a function of the configurational coordinate Q, and -( 1/2M)V2 is the nuclear kinetic energy. For two coupled states, it is given by

with the diagonal elements Hiof the total Hamiltonian as given in eq 6. The split-operator method developed by Feit et al. is used to calculate Both the configurational coordinate Q and the time are represented by a grid with points separated by AQ and Ar, respectively. For one surface, the time-dependent wave function $(Q,t+At) is obtained from 4(Q,t) with $i(t).9320*21

#(?+At) = exp(

(g ) V 2 ) exp(-iAtV) exp((3 - V214(Q,t) +

Wexler et al. operator Pis more intricate. The exponential operators must be given in terms of the potentials that diagonalize the potential matrix in the total Hamiltonian, eq 7,Le., in terms of the adiabatic potentials V, and V,. These potentials are calculated from the diabatic potentials VI and V, and the coupling V12: V, = c]Vl Vb

+ c2V2 = f/z[(Vl$. V2) - ((VI - V2)'+

+ V2) + ((VI - V2)'+

= CzV, - C l V 2 = X{(V1

4V12)'/~) (10) 4V,2)'/') (11)

Exact numerical solutions to the time-dependent Schriidinger equation are obtained by using eq 9. The coupling between the potential surfaces is calculated to all orders in eq 9 in contrast to the first-order perturbation expressions in eqs 1 and 2. From eq 9, it is obvious that I$](?) and $ ~ ~ (are t ) mixed (formally via the off-diagonal matrix elements V12)at each time step. Details of the computer implementation of eq 9 are given in the literature.I2-l9 It is interesting to note the different roles of d and p. P,the momentum operator, transfers the wave function amplitude 4i among grid points along Q at each time step but does not transfer the amplitude between the diabatic parent states 1 and 2. These changes are easily monitored by looking at>he wave packet &(t) after every time step. On the other hand, V, the potential energy operator, transfers the amplitude between the electronic states at each time step but does not couple grid points along Q. The amplitude transfer between the diabatic potentials can be followed for 4](t) and after by calculating the norms (bi(t)lq$(t)) every time step. The norms are a quantitative measurement for the amount of population change between the two states. Both operators affect the total overlap (414(t))in eqs 3 and 4 and therefore the spectra. Only the calculation which simultaneously involves both coupled states according to eq 9 gives the correct total overlap. All calculations involving only one surface lack the contribution of the population change between the states to the total dynamics due to nonadiabatic transitions or "surface hopping". In the time-dependent picture, it is straightforward to see how these population changes, direct manifestations of the breakdown of the Born4ppenheimer approximation, occur and what their influences are on the absorption spectrum. Experimental absorption spectra often show progressions in many normal modes. In the framework of timedependent theory, it is easy to calculate a spectrum involving many uncoupled normal coordinates by using a series of one-dimensional propagations. In the absence of coupling terms between the normal coordinates, the total autocorrelation in a system with k coordinates is given by To calculate the emission spectra of molecules in condensed media at low temperature, the initial wave packet is the vibrational eigenfunction corresponding to the lowest eigenvalue of the coupled excited-statesurfaces. The eigenvalue can be found by calculating the absorption spectrum. Once the eigenvalue is known, the eigenfunction is calculated by using equation 1 3.9 qi denotes

O[(Ar)3] = PPPI#J(QJ)+ O[(At)3] (8)

The generalization of this equation to the case of two Fupled potentials requires that the exponential operators and Vin eq 8 be replaced by 2 X 2 matrices operating simultaneously on 4i(QJ)and MQJ)

( " )( 2)("" " )( '0I

P,

p2

2(Q4

+ Ol(Al)3]

(9)

The kinetic energy operator P is independent for and d2in the diabatic basis, Le., its matrix is diagonal. The potential energy

the eigenfunction corresponding to the eigenvalue E,, 4(t) is the time-dependent (propagating) wave function, and w ( t ) is a Hanning window function. The eigenfunction is multiplied by the transition dipole moment and propagated on the ground-state potential surface by using eq 8. The emission spectrum is calculated by using eq 4. For coupled potentials, each eigenfunction \kiis an array with two components corresponding to the projection of the exact eigenstate onto the basis states 1 and 2 with their respective spins. These projections will be referred to as "the parts of the eigenfunction associated with states one and two". Details and examples of the calculation of emission spectra from two coupled excited states are discussed later.

Vibronic Structure Induced in Spin-Forbidden Transitions

The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8759 1.o

0

160

2b0

300

400

Time [fs] 0.0

0.2

0.4

1

Configurational Coordinate Q [A]

"............ .."

,a'

0.15

..I'

6

5 -04 -02

00 02 04 06 Configurational Coordlnate Q [A1

Figure 1. Potential energy surfaces for the calculations of intensity borrowing. The diabatic and adiabatic surfaces are shown as bold and dotted lines, respectively. Labels 1 and 2 denote diabatic surfaces with small (or zero) and large displacements, respectively. (a) The highest energy diabatic potential surface 1 represents the state to or from which the electronic transition is forbidden. (b) The lowest energy diabatic potential surface 1 represents the state to or from which the electronic transition is forbidden. This state is undisplaced along the normal coordinate from the ground state.

3. Electronic Spectroscopy Involving Excited-State Potential Surface Coupling A. Intensity Borrowing. Induced Oscillator Strength in A b sorptioa Spectra. In the time-domain picture, amplitude transfer between surfaces due to coupling can cause a spin-forbidden transition to gain intensity in the electronic spectrum. A forbidden transition means that the initial wave packet is multiplied by zero (the transition dipole moment) and thus does not have any amplitude on the potential surface of the state (called "state 1" in this paper) with spin multiplicity different from that of the ground state. However, for the spin-allowed transition, the wave packet is multiplied by the non-zero transition dipole moment, and the wave packet is transferred vertically from the ground state onto the potential surface (called "state 2"). When spin-orbit coupling between the two states is non-zero, amplitude is transferred from state 2 to state 1. The quantitative relationship between amplitude transfer in the time domain and observed intensity in the spectrum in the frequency domain depends upon how fast the amplitude transfer occurs and how much amplitude is transferred. These two aspects are interrelated and depend in part on the coupling strength and the shapes and positions of the potential surfaces. Amplitude transfer occurs immediately after the transition and begins with the first time step in the calculation. Our goal is to calculate quantum mechanically the exact spectral intensities and relate them to spin-orbit coupling between the surfaces. Amplitude transfer between the surfaces is calculated from the populations of the two surfaces as a function of time. The pop ulation Pi (I'= 1,2) is defined as the norm of the time-dependent wave function &(r): PI = (4J,(~)l4Jl(~)) (14) At zero time, the population of state 1 is zero and that of state 2 is one.

200

400

600

Coupling v,? ~cm-']

Figure 2. Top: Time-dependent populations of the diabatic potential surfaces in Figure la. Three coupling strengths are shown: VI,= 0 cm-' (dashed lines), VI,= 400 cm-I (dotted lines), and VI,= 700 cm-' (solid lines). Bottom: Ratio of the intensities of forbidden to allowed absorp tion bands (Figure 3) as a function of the coupling strength. The points denote ratios determined from calculated spectra with r = 15 cm-l. The dashed curve is a least-squares fit to a parabola. The ratio of the average populations of states 1 and 2 at three coupling strengths is given by the open squares. A pair of coupled potential energy surfaces that will be used to illustrate intensity borrowing are shown in Figure la.22 Excited-state diabatic potential surfaces 1 and 2 have an identical vibrational fr uency of 600 cm-I, their minima are displaced by 0.01 and 0.19 along the configurational coordinate Q, and the energy separation of the excited-state minima is 4800 cm-'. The ground-state potential surface has a vibrational frequency of 700 cm-I. The energy of the electronicorigin transition is 11 200 cm-l. These parameters of the potential surfaces were chosen to represent approximately the region of the absorption bands of the linear Ni022- chromophore in K2Ni02,with Q representing the totally symmetric breathing mode.23 In this compound, the spin-forbidden band adjacent to a spin-allowed band gains s i g nificant intensity. For illustrative purposes, the energy of the potential minimum for the triplet excited state is shifted to lower energy than that observed in the experiment in order to separate the allowed and forbidden absorption bands. A fit to the experimental spectrum for thii compound will be presented in section 4. The norm of the wave functions as a function of time is shown in Figure 2a for three different couplings V12.The transfer of wave-packet amplitude starts at the first time step of the calculation. Two trends are observed in Figure 2a. First, the "average population" of state 1 increases with increasing coupling. The higher population leads to higher intensity for the transition to state 1 in the absorption spectrum. Second, the initial transfer of wave-packet amplitude to state 1 during the fmt few time steps increases as the coupling increases. This trend is apparent from the magnitude of the first peak of the population at 6 fs. Therefore, the intensity borrowing increases as a function of coupling even in completely unresolved, broad-band spectra which are determined by the short-time wave-packet dynamics. The calculated absorption spectra for the potential surfaces in Figure la are shown in Figure 3. All of the calculations are done with dipole moments of 0 and 1 for the transitions to states 1 and 2, respectively. In the uncoupled case (VI, = 0 cm-I), only the allowed transition is seen in the spectrum. At moderate coupling strength (V12= 400 cm-I), the forbidden transition appears as a series of lines on the high-energy side of the allowed transition

1

3n,/1c,+

8760 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 I

I

I

I

I

12

14

16

,

IS1

20

Wakenumber [cm.’] x10-

Figure 3. Absorption spectra calculated for the potentials in Figure l a with damping factors r of 15 cm-’ (solid lines) and 250 cm-I (dotted lines) at three different coupling strengths. The coupling constants are 0, 400, and 700 cm-l from top to bottom. Vibronic bands that gain intensity with increasing coupling are denoted by arrows.

with enough intensity to be experimentally observable. The lines become more intense at a coupling strength V , , of 700 cm-I. The ratios of the intensities of the forbidden and allowed absorption bands are plotted in Figure 2b as a function of VI*. They show a nonlinear intensity increase that can be well-described by a second-order polynomial. A dependence of the intensity ratio on the square of the spin-orbit coupling parameter is predicted from simple perturbation in agreement with our timedependent calculations. The ratios P l / P 2 of the “average populations” of states 1 and 2 for the three couplings shown in Figure 2a are included in Figure 2b for comparison. The average populations are calculated as the mean values of the populations in the time range shown in Figure 2a. They almost perfectly reproduce the dependence of the spectroscopicintensity ratios on coupling strength, indicating that wave-packet amplitude transfer between states 1 and 2 is indeed the determining factor for intensity borrowing. The intensity borrowing is also affected by the energy separation between the coupled surfaces. When the coupling between the surfaces is constant, the general trend is an increase in the intensity of the forbidden band as the energy separation decreases. Amplitude transfer from the allowed to the forbidden surface is responsible for the induced intensity as was discussed above. Emission Spectra and Excited-State Eigenfunctlons. To calculate the emission spectrum resulting from a transition from the ground vibrational state of the coupled excited electronic states, the eigenfunction of the lowest vibrational level is multiplied by the transition dipole moment and propagated on the ground-state potential surface. The procedure for calculating the emission spectrum therefore consists of calculating the eigenvalues of the coupled excited states (for example, by calculating the absorption spectrum), calculating the eigenfunction of the lowest vibrational level of the excited state by using eq 13, multiplying the eigenfunction by the transition dipole moment, and propagating the wave packet on the ground electronic state potential energy surface. There are several nonintuitive features involved in this procedure that are discussed below. The eigenfunction consists of a part from surface 1 and a part from surface 2. In our example (Figure lb), the lowest eigenvalue is near the minimum of surface 1 and well below the minimum energy of surface 2. When VI, = 0, the eigenfunction is of course completely associated with surface 1 and the component from surface 2 is zero. However, when the coupling is non-zero, the complete eigenfunction has a nonzero component from surface 2 in addition to that from surface 1, even though the minimum energy of surface 2 is greater than the eigenvalue. The coupling-induced relaxed emission intensity from spinforbidden surface 1 arises from the component of the lowest energy eigenfunction associated with surface 2. To calculate the emission spectrum, the component of the eigenfunction from surface 2 is multiplied by the non-zero transition dipole moment corresponding

Wexler et al. to the spin-allowed transition from surface 2, and the wave packet is propagated on the ground-state surface. (The component of the eigenfunction from surface 1 is multiplied by zero, the transition dipole moment for the spin-forbidden transition.) Only the component from surface 2 of the lowest energy eigenfunction contributes to the emission intensity. It depends on the coupling between the excited-state surfaces and vanishes when the coupling is zero. B. Vibronic Structure Induced by Coupling in Absorption and Emission Spectra. In this section, we will focus on the intensity distributions in vibronic progressions in electronic spectra arising from coupled electronic excited states. Our goal is to show how the time-dependent point of view provides both a physical picture and an exact quantum mechanical calculation of the full vibronic part of the problem. Absorption Spectra. As a starting point in interpreting the physical meaning of the effects of electronic state coupling on the vibronic structure, the problem can artificially be separated into two categories. First, in the limit where the adiabatic surfaces are a good approximation, the effects of the coupling on the “shapes” of the adiabatic potential surfaces provide insight into the origin of the vibronic structure. Second, the time-dependent viewpoint shows how the coupling affects the transfer of amplitude between the two diabatic surfaces. Consider first the shapes of the adiabatic surfaces. As a pedagogical starting point, consider two harmonic diabatic surfaces that are coupled. In general, as the coupling between the surfaces increases, the adiabatic surfaces become less and less harmonic. The effective ’force constant”, or more precisely the curvature at the minima of the surfaces, will change. In addition, both the position and the energy of the minima will change. For example, as the coupling between the surfaces shown in Figure 1b increases, the lowest adiabatic surface becomes asymmetric, and most importantly, the position of the minimum shifts along the normal coordinate (in a direction toward the position of the minimum of the upper diabatic surface.) This shift results in a ‘displacement” of the minimum along the normal coordinate and will also cause a vibronic progression to appear instead of the single band expected without coupling. Consider next the effects of the transfer of amplitude between the two diabatic surfaces. The wave packet exhibits dynamics on both coupled surfaces, and thus, the spectra will show vibronic features from both surfaces. Because amplitude is transferred between surfaces and the wave packet is constantly changing, the vibronic structure in the spectrum will be different from that caused by placing the initial wave packet onto one surface at a time and adding the spectra obtained for both surfaces. The wave packet that develops amplitude on a given surface due to coupling will be different from a wave packet 4(t=O) which is placed directly on the surface. From just this qualitative point of view, it is to be expected that the autocorrelation functions and hence the spectra will be different. The quantitatively calculated spectra resulting from a transition from a harmonic ground-state surface to coupled excited-state potential surfaces illustrate the qualitative discussion above. When the potential surfaces are nested, i.e., when the minima of the diabatic surfaces are both at Q = 0, the exact calculations show that a significant vibronic progression is not induced when physically meaningful differences in the force constants of the potentials (- 10%) are used and when the state separation energy (the separation of the minima of the diabatic surfaces) is at least four vibrational quanta. Very pronounced vibronic intensity changes occur at the energies where the diabatic potential surfaces cross, but the crossings can only occur at energies much higher than those of the intense vibronic transitions unless there are very large differences between the force constants describing the potential and/or the energy separation between the surfaces is on the order of one vibrational quantum. Because these situations are not very common, we will not analyze them further. The most commonly encountered disposition of potential surfaces is that illustrated by the examples in Figure 1 where the minima are displaced both in energy and in normal coordinate

Vibronic Structure Induced in Spin-Forbidden Transitions

The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8761

\ . .-e

t

5

I

I

10

I5

-0.3

1

Wavenumber [cm”] ~10.’

Figure 4. Absorption spectra showing the vibronic progression induced in the forbidden transition (lowest energy bands) by the coupling. The spectra were calculated with the potential energy surfaces in Figure lb. The coupling constants are 0, 1000, and 3000 cm-I from top to bottom.

space from each other. In order to best illustrate the effects of coupling on vibronic progressions, the spectra resulting from transitions to the surfaces in Figure l b will be discussed. The absorption spectra are shown in Figure 4 as a function of the magnitude of the coupling between the surfaces. The relative intensities of the vibronic features from each individual state change as a function of coupling. The vibronic structure is not simply a progression multiplied by a constant related to the coupling strength. In the “forbidden” state, for example, significant intensity appears in the u = 1 quantum line euen though the diabatic potential surface for this state is undisplaced and there is noforce constant change between the ground and excited states. Likewise, the relative intensities between the vibronic peaks in the allowed transition change as the coupling is changed. The numerical values of the vibronic intensities can readily be calculated by using the split-operator method. However, the trends in the relative intensities in a progression do not necessarily follow a simple pattern. The vibronic intensities must be calculated on a case by case basis. Examples of the applications of these calculations to the spectra of metal complexes are discussed in detail later. Emission Spectra. The trends in the intensity distributions in the vibronic structure induced by coupling between excited-state potential surfaces can be interpreted from the trends in the eigenfunction of the lowest energy vibrational level of the coupled excited states. As discussed previously, the initial wave packet consists of two parts: one from the part of the eigenfunction associated with diabatic surface 1 multiplied by the transition dipole between surface 1 and the ground state, and the second part from the part of the eigenfunction associated with diabatic surface 2 multiplied by the transition dipole between surface 2 and the ground state. In the examples discussed in this paper, the transition between surface 1 and the ground state is spinforbidden, and thus, the transition dipole is zero. The parts of the eigenfunction associated with surface 1 and surface 2 for the lowest vibrational level of the coupled excited states illustrated in Figure 1b are shown in Figure 5 . Eigenfunctions are shown for four values of the coupling including zero. (When the coupling is zero, the eigenfunction is associated entirely with surface 1.) The signs of the eigenfunctions are shown; the square of the function is the probability. The results are not immediately intuitive. For small values of the coupling, the eigenfunction associated with surface 2 has most of its probability at a position near the minimum of surface l! As the coupling increases, the probability moves away from the minimum of surface 1 in the direction of the minimum of surface 2. The spectra calculated by propagating the above functions on a harmonic ground-state potential are shown in Figure 6. For zero coupling, there is no emission spectrum because the transition is dipole-forbidden. As the coupling increases, the intensities of the vibronic side bands relative to that of the origin increase. A

I

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-0.2

-0.1

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QIAl Figure 5. Eigenfunctions corresponding to the lowest energy eigenvalue of the coupled states illustrated in Figure lb. The sign of the part of the eigenfunction associated with state 1 is positive; the sign for the part associated with state 2 is negative. The eigenfunctions arising from couplings of 0,500, 1000, and 3000 cm-’ are shown. When the coupling is zero, the part of the eigenfunction from state 2 is zero.

0 8.0

8.5

9.0

9.5

10.0

Energy [cm-’1x l o 3

Figure 6. Calculated emission spectra corresponding to the potential surfaces shown in Figure Ib. The coupling constants are (a) 500, (b) 1000, and (c) 3000 cm-l. The vibronic progression in the totally symmetric mode induced by the coupling is evident. The transition is forbidden when the coupling is zero.

simple physical picture for explaining the increasing intensity of the side bands to that of the origin is that the wave packet has increasing probability a t a position away from the minimum of the ground-state surface, giving rise to more rapid decreases in (r##(t)) and hence to a broader spectrum containing more vibronic features. (In the time-independent picture, the initial eigenfunction appears to be more and more displaced, leading to better overlap with higher quanta vibrational wave functions of the ground electronic state.) The net results of increasing the excited-state potential surface coupling are to make the spin-forbidden transition become allowed and to induce a vibronic progression in the emission from an undisplaced potential surface. The magnitudes of these effects depend not only on the coupling but also on the relative placements of the potential surfaces. Although the trends can be understood in terms of the physical pictures discussed above, case by case quantitative calculations must be carried out in order to understand the details in the spectra of specific molecules. A quantitative analysis of the resolved vibronic structure in the absorption and

8762 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992

Wexler et al.

3ng

of the excited state is lower than that of the singlet excited state because an orbital with more antibonding character is occupied in this state. The coupling constant between the two surfaces is on the order of the spin-orbit coupling constant determined to be 460 cm-' from the ligand field calculati0ns.2~The numerical range for all the parameters is restricted by the spectroscopic data or by the ligand field calculations. The two least restricted parameters to be refined are the positions of the diabatic potential minima along the a l normal coordinate for the two excited states. The calculated spectrum in the region of the /IC,+ bands is compared to the experimental data in Figure 7. h e agreement is good; the relative intensities, the shapes, and the energies of the bands are well-reproduced by the calculation. Changing the coupling strength changes the intensity ratio, but it also affects the vibronic structure, leading to an incorrect calculation of both the band positions and the intensity distribution in the ICg'progression. Similar effects are observed by changing the intensity of the transition dipole ratios for the spin-forbidden and spin-allowed transitions. They are not simple scaling factors but have more complicated effects on the calculated intensities of both bands. For the calculation of the spectrum, a ratio of 0.06:l for the transition moments of the forbidden and allowed bands was found to reproduce the experimental spectrum best. The spectrum of K2Ni02shown in Figure 7 is an experimental example for absorption intensity caused by wave-packet amplitude transfer between coupled potential surfaces, as discussed in section 3. Similar spectra are reported for Ni2+doped in chloride and bromide lattices, and the same intensity mechanism applies to those systems as well.27 The vibrational frequencies for the diabatic potential surfaces leading to the calculated spectrum in Figure 7 are 733, 590,and 500 cm-' for the ground state and the singlet and triplet excited states, respectively. The coupling constant in the best fit is 530 cm-I. The energy of the electronic origin transition -c ), corresponding to the mini"of the potential surface of the tripfet excited state, was determined to be 13690 cm-' from the experimental spectrum. The damping factor I' for the calculated spectrum is 168 cm-'. The minima of the diabatic surfaces are separated by an energy of 2660 cm-'. The displacements of the minima along the totally symmetrical normal coordinate are AQ,, = 0.074 A and AQ,,, = 0.194A for the singlet and triplet excited states, respectively. From these displacements, the changes of the Ni-O bond length in the two excited states are calculated as

3n

I

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20 Wavenumber [cm'l] ~10.'

14

16

18

I

I

22

Figure 7. Experimental (top and bottom) and calculated (middle) absorption spectra for K2Ni02. The polarizations of the single-crystal spectra (measured at 22 K)are indicated. The bands corresponding to transitions to excited states 1 and 2 (Figure l a ) are labeled on the calculated spectrum. The baseline used in the calculation is given as a dotted line.

emission spectra of metal complexes is given in the next section.

4. Calculation of Vibronic Intensities in the Spectra of Metal Complexes Induced by Spin-Orbit Coupling A. Absorption Spectrum of K2Ni02. In this section, we present and interpret the low-temperature single-crystal absorption spectrum of K2Ni02,a new type of nickel oxide containing linear Ni022- chromophore^.^^,^^ This compound is one of a small number of examples where Ni2+is coordinated by only two ligands. The crystal structure, vibrational spectroscopy, and a low-resolution absorption s p e c " in the visible and near-infrared spectral region have been r e p ~ r t e d . Ligand ~ ~ , ~ ~field calculations in Dmh symmetry were successfully used to assign the symmetries of the excited electronic states observed in the absorption spectrum.23 We use these assignments and calculate the relative intensities and vibronic structure of transitions to coupled electronic excited states in this compound. The polarized absorption spectra of K2Ni02at 22 K are shown in Figure 7. The spectra were measured on a single crystal with approximate dimensions of 200 pm X 130 pm X 100 pm. The compound is very hygroscopic, and therefore, the samples were enclosed in a quartz capillary. The instrument used to measure the absorption s p t r a of small single crystals has been described elsewhere.25 A broad band with its maximum at 16000cm-l and polarized parallel to the Ni-O axis dominates the spectrum in the visible range. Vibronic structure is observed on its high-energy side, with a spacing of 550 cm-' between the first two bands at 16850 and 17 400 cm-I and a spacing of 685 f 10 cm-l between the subsequent members of the progression. From the li and field calc~lations,2~ the broad band is assigned as the 3$ spin-allowed transition. The progression on its high-energy side is assigned as the spin-forbidden IC,' transition. This assignment is not unique: small changes in the ligand field parameters can change the energetic order of three singlet excited states in this region of the spectrum. A weak spin-forbidden transition is observed at 13 700 cm-I in the opposite polarization and assigned as IA,. Our low-temperature spectra show the different transitions at high resolution, allowing the energies of the excited states to be determined more precisely than from the room-temperature spectra.23 The general features of the spectra, such as band polarizations and band positions, are identical in the two sets of measurements. The intensity of the 3& ICg+ transition is indicative of intensity borrowing,26as illustrated in section 3. A onedimensional model with three potential surfaces similar to those shown in Figure la is used. We assume the configurational coordinate to be the totally symmetric a l amode of the NiOz2-molecule. The ground-state vibrational frequency for this made is 732.5 cm-1.23 The excited-state vibrational frequency for the 'Cy+ electronic state is determined to be between 550 and 680 cm- , the energy intervals observed in the progression. The vibrational frequency

-

jn8

-

3ns

-

- 'n,

3n

Excited-state distortions of 0.06 and 0.14 A are calculated for the singlet and triplet excited states, respectively. The bond length changes by 8% and 4% for the two excited states compared to the ground state. An antibonding orbital becomes populated in the triplet excited state; therefore, each Ni-O bond is lengthened by 0.14A in this excited state. The sign of the distortion in the triplet state determines the sign of the bond length change in the sin let excited state: in K2Ni02,the bonds are elongated by 0.06 in this excited state. A spectrum calculated with AQ, (IC,+)= 4.08 A, corresponding to a shortening of the Ni-O kond in the excited state, is dramatically different from the experimental data. Any model calculation neglecting the coupling between these excited states is not able to discriminate between a bond elongation or contraction in the intraconfigurational IC,' transition. The other interesting feature in the experimental absorption spectrum is the nonconstant energy interval separating the members of the progression forming the 'Cg+band. The first two members of the calculated spectrum in Figure 7 are separated by 604 cm-I; the following two energy differences are 650 and 660 cm-I, respectively. The same trend as in the experimental spectra is obtained from the calculation, although the energy differences do not match perfectly, most likely due to the simple model taking into account only one normal coordinate. These nonconstant splittings are another manifestation of coupled potential surfaces. A simple qualitative explanation is evident from the adiabatic surfaces in Figure 1 b, which are very similar to those

x

'n,

-

The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8763

Vibronic Structure Induced in Spin-Forbidden Transitions

TABLE I: Calculated and Observed Spectroscopic Features intensity ratio“ metal calcd obsd Ti2+ 0.044 C

VI,,cm-’ 85 130 150 220

allowed band max calcd (obsd)bcm-I

allowed band width calcd (obsd)bcm-I 1075 (1100 100)

*

** *

9260 (9260 20) 0.12 f 0.05 1 1 190 (11 100 i 50) 1200 (1400 100) 0.121 1850 (2150 100) 0.07 0.03 16400 (16300 50) Cr3+ 0.091 Mn4+ 0.006 0.004 0.001 21 614 (21600)d 2100 (4 “Ratio of the u = 1 to u = 0 peak intensities in the algvibronic band. bThe experimental values define the potential surfaces used in the calculations (see text). CTheintensity is lower than the signal-to-noiseratio. dThe experimental values are not precisely reported in ref 32. v3+

*

for K2Ni02. The upper adiabatic surface (representing the I&+ excited state) is harmonic and very similar to the diabatic potenbal near its minimum, but a t energies close to and higher than the crossing point of the diabatic surfaces, it becomes narrower than a harmonic potential, leading to an increased splitting of its levels. This simple qualitative argument based on the adiabatic surfaces can rationalize the observed trend in the experimental spectrum. The observed transition intensities of the spin-allowed and spin-forbidden bands in K2Ni02are quantitatively explained with coupled one-dimensional excited-state surfaces. A model taking into account more than one dimension would be needed to refine the agreement between the observed and calculated band shapes of the triplet band. In addition, including distortions along other normal modes in the triplet state would lead to an unresolved triplet band, as is observed. In this work, our goal was to analyze the intensitiesof the absorption transitions to the coupled excited states and the vibronic structure of the singlet band. For this purpose, a one-dimensional model with a minimum number of parameters is sufficient. B. Emission Spectra of Ti2+and VM in Chloride Lattices.8*28 In this section, the electronic spectra of Ti2+ and V3+ ions in octahedral chloride lattices are analyzed. In this pair of d2 metal ions, highly resolved spin-forbidden emission spectra from the lowest energy singlet excited state to the triplet ground state and spin-allowed absorption spectra are observed. The lowest energy spin-forbidden transition is a “pure spin-flip”; Le., the orbital part of the wave function is unchanged in the transition. Thus, no bond length changes are expected. However, the emission spectrum of the V3+ ion shows induced vibronic intensity in the algmode.!’ The spectrum of the Ti2+ion does not contain induced intensity.28 The induced intensity in the former ion is quantitatively calculated in terms of the spin-orbit coupling between the forbidden singlet and allowed triplet excited states. The absence of induced intensity in the Ti2+ion is a result of the smaller spin-orbit coupling constant and larger energy separation of the singlet and triplet excited states. These specific examples were chosen because the vibronic structure is well-resolved and the energies of the excited states are accurately measured.8*28 The emission and absorption spectra of V3+and Ti2+ions doped in Cs2NaYCl, and MgC12 have been reported and assigned.8*28 The emission spectra are well-resolved at low temperature. Both spectra contain vibronic bands corresponding to undisplaced ungerade promoting modes. The spectrum of the V3+ ion contains a short progression in the algmetal-chloride stretching mode. This mode is absent in the spectrum of the Ti2+ ion. The potential surfaces similar to those in Figure lb, appropriate for calculating the spectra of these ions, are defined by the well-resolved spectra. The energy of the undisplaced potential surface corresponding to the singlet state is obtained from the emission spectrum, and its excited-state vibrational frequency is unchanged from that of the ground state because the transition involves only a change in the spin. The energy of the displaced spin-allowed triplet state is obtained from the absorption spectrum. The vibrational frequency in this excited state is estimated to be 80%of the ground-state value, and the displacement is calculated from the bandwidth. The spin-orbit coupling constants {for V3+ and Ti2+,obtained from ligand field calculations, are 130 and 93 cm-I, respectively. The calculated and experimental values of the spectroscopic observables are given in Table I. The feature in the emission spectrum that is the focus of these calculations is the induced intensity in the a,* V-Cl stretching

-

8

615 710 715 Energy [cm.’] a I O ’

8;O

8.0

8.5 9.0 9.5 Energy [cm.’~x 10”

10.0

13.0

13.5

15.0

6.0

3 0.4 0.0

0.8

0.0 14.0

14.5

Energy Icni’I x I O ’

140

145

I i O 15: Energy lcm ‘I x I O

160

Figure 8. Calculated intensitiesof the electronicorigin and the u = 1 and higher al, vibronic bands in the emission spectra of the d2 Ti2’ and V3+ ions and the d3 Cr3+and Mn4+ions in the octahedral halide lattices discussed in the text. The intensities are induced by spin-orbit coupling. The potential surfaces used in the calculations are defined by the following values. Ti2+(top): ground-state potential vibrational frequency (hw,, ) = 260 cm-I, forbidden excited-state vibrational frequency (hwr,,, ) = 260 cm-’, allowed excited-state vibrational frequency (huallexsI) = 200 cm-I, VI,= 85 cm-I, AE,,, = 800 cm-I, AQ,, = 0.21 A. V3+ (upper middle): hug,,, = 300 cm-I, hwIorcxsI= 291 cm-I, hudI~x,I = 215 cm-I, VI,= 130 cm-I, AE,,, = 170 cm-’, AQ,, = 0.235 A. Cr3+(lower middle): husrrt= 568 cm-I, hwforexst= 568 cm-I, hwauex,I=510cm-l, V 1 2 =150~m-~,AE,,= ll00cm-’,AQ,, -0.125 A. Mn4+ (bottom): huprsI= 595 cm-l, hufwcxsI = 595 cm-l, h ~ = 490 cm-’, VI,= 220 cm-I, AE,,,,,= 7640 cm-l, AQ,,,,,, = 0.15A. Other relevant spectroscopic properties are given in Table I.

mode. The experimentally observed intensity ratio of the band corresponding to one quantum of this vibration to that of the appropriate band containing zero quanta of this mode is 0.12 f 0.05:1. The experimental uncertainty arises because of the overlap of this band with others in the spectrum. The other bands, primarily undisplaced ungerade enabling modes, are not part of the calculations discussed below. The results of fitting the experimentally determined induced intensity ratio of 0.12:l and the other spectroscopic features discussed above are given in Table I and are illustrated in Figure 8. The excited-state spin-orbit coupling constant that is required for the fit, 130 cm-l, is the same as that reported in the literature for V3+.8 Both the width and the maximum of the broad spinallowed absorption band are accurately calculated. As shown in Figure 8, the calculated intensity ratio for the first two members of the a l e mode is 0.12:1, the same as that observed in the experimental spectrum. In addition, Figure 8 shows that the calculated intensity of the u = 2 band is appreciable (0.051). In

~

,

Wexler et al.

8764 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 fact, this band also appears in the experimental spectrum but its intensity cannot be measured accurately because of severe overlap with other bands. The experimental spectroscopic observables place severe constraints on the calculation. In order to be an accurate representation of the excited states of these metal ions, the potential surfaces must not only account for the induced vibronic intensity but also must yield the band energies and bandwidths of both the allowed and forbidden transitions observed in the experimental emission and absorption spectra. In fact, in order to simultaneously satisfy all of these constraints,the values used to construct potential surfaces such as those in Figure 1 are determined to within a few percent. As an example of the effect of small changes, when the spin-orbit coupling constant is changed through a range of *30 cm-I and the energy separation between the diabatic surfaces is changed by f40 cm-I, the calculated intensity ratio can be made to match the observed intensity ratio of 0.12:l. (If the spin-orbit constant is increased, the energy separation must also be increased, and vice versa.) With such changes, however, both the calculated intensity of the forbidden absorption band and the position of the allowed absorption band deviate significantly from those in the experimental spectra. The emission spectrum of Ti2+does not contain observable induced intensity in the alamode. This result supports the theory presented here because the spin-orbit coupling constant is smaller for Ti2+than for V3+,and thus, the induced intensity should be smaller as discussed in section 3. In order to more quantitatively test the theory, the coupled potential surfaces must accurately reproduce the other spectroscopic features. The quantities defining the potential surfaces and the results of the calculation are given in Table I. Because the spin-forbidden emission is observed, the spin-orbit coupling constant must be non-zero and the induced vibronic intensity will be small but non-zero. From the experimental emission spectrum, we estimate that the upper limit of the ratio of the induced intensity to the zero quantum band is 0.05:l. From the calculations, the spin-orbit coupling between the two excited states is 85 cm-I, in good agreement with the literature value of 93 cm-1.28 The Ti2+and V3+ ions in chloride lattices provide a good test of the theory of induced vibronic intensity. The presence of the induced progression in the spectrum of V3+ and the absence in that of Ti2+is quantitatively explained by the different spin-orbit coupling constants and different excited-state energies. c. I!dssiioa spectra of w and Mn*c in FhKwide Iattice8.2’32 The electronic spectra of the d3 metal ions Cr3+ and Mn4+ in octahedral fluoride lattices contain the highly resolved vibronic structure in the spin-forbidden emission spectra. The transition is a pure spin-flip. Because the orbital part of the wave function is unchanged, no bond length changes are expected. However, the emission spectra show an induced vibronic intensity in the alg mode. The induced intensity is quantitatively calculated in terms of the spin-orbit coupling between the forbidden doublet and allowed quartet excited states. These specific examples were chosen because the vibronic structure is well-resolved and the energies of the excited states are accurately measured.2F32 The emission spectrum of Cr3+doped in K2NaAlF6contains a short progression in the aIgCr-F stretching mode together with vibronic bands corresponding to undisplaced ungerade promoting modes.29 The feature in the emission spectrum, which is the focus of these calculations,is the induced intensity in the alamode. The experimentally observed intensity of the band corresponding to one quantum of this vibration to that of the appropriate band containing zero quanta of this mode is 0.07 f 0.03:l. The experimental uncertainty arises because of the overlap of this band with others in the spectrum. The results of fitting the experimentally determined induced intensity ratio and the other spectroscopic features discussed above are given in Table I and are illustrated in Figure 8. The 150-cm-l excited-state spin-orbit coupling constant that is required for the fit compares well with the value of 172 cm-I that was determined from a ligand field calculation. In addition, both the width and the energy of the maximum of the broad spin-allowed absorption

band of K2NaCrF, are accurately calculated. The emission spectrum of the d3 Mn4+ion in Cs2MnF6is the best resolved of all of the spectra treated here.32 The spin-forbidden emission contains a very weak band arising from one quantum of the ala Cr-F stretching mode, together with vibronic bands corresponding to undisplaced ungerade promoting modes. The experimentally observed intensity of the band corresponding to one quantum of this vibration to that of the appropriate band containing zero quanta of this mode is 0.004f 0.001:l. This weak band is discernible in the spectrum because it is not severely overlapped by other bands and because it occurs in a region where the baseline is flat.j2 The results of fitting the experimentally determined induced intensity ratio and the other spectroscopic features discussed above are given in Table I and illustrated in Figure 8. It is interesting to note that the spin-orbit coupling constant, 220 cm-I, is the largest for the metals that have been discussed in this paper but that the induced intensity is one of the smallest. The major reason for the small value of the induced intensity in one quantum of alais the relatively large-energy separation of 7640 cm-I between the forbidden potential surface and the allowed surface. 5. Summary Exact quantum mechanical calculations of the spectroscopic effects of excited electronic state coupling along one vibrational normal coordinate were carried out. The integrated intensity induced in a forbidden transition (intensity borrowing) and the relative intensities of vibronic peaks in emission and absorption spectra were calculated. The theory was applied to the intensities and vibronic structure in forbidden bands in the absorption spectrum of K2Ni02and to the intensities of vibronic bands in the spin-forbidden emission spectra of d2and d3 ions in octahedral halide lattices. Both the bandwidths and peak maxima of the allowed transitions in the experimental absorption spectra and the energy of the forbidden transition in the absorption or emission spectra define the excited-state diabatic potential surfaces within a very narrow range of energies, displacements, and force constants. The defined surfaces and known values of spin-orbit coupling were used to accurately calculate the relative intensities of induced vibronic progressions.

Acknowledgment. This work was made possible by a grant from the National Science Foundation (CHE91-06471). We thank Dr.J. Gauss and Prof. E. J. Heller for their implementation of the propagation algorithm and Prof. M. A. Hitchman for providing the crystal of K2Ni02. References and Notes (1) Zink, J. I.; Shin, K. S.K. In Advances in Photochemistry; Volman, D. H., Hammond, G. S., Neckers, D. C., Eds.; Wiley: New York, 1991; Vol. 16, p 119. (2) Ballhausen, C. J. Introduction to Ligand Field Theory; McGraw-Hill: New York, 1962. (3) Schlaefer, H. L.; Gliemann, G. Basic Principles o j Ligand Field Theory; Wiley: London, 1969; p 96. (4) Jsrgensen, C. K.Acta Chem. Scand. 1955, 9, 1362. (5) Kirk, A. D. Coord. Chem. Rev. 1981,39,225 and references therein. (6) Flint, C. D.; Matthews, A. P. Inorg. Chem. 1975, 14, 1008. (7) Ferguson, J. frog. Inorg. Chem. 1970, 12, 159. (8) Reber, C.; Giidel, H. U. J . Luminesc. 1988.42, I . Reber, C.; Guedel, H. U. In Photochemistry and Photophysics of Coordination Compounds; Yersin. H., Vogler, A,, Eds.; Springer: Berlin, 1987; p 17. (9) Feit, M. D.; Fleck, J. A.; Steiger, A. J. Compur. Phys. 1982, 47,412. (IO) Heller, E. J. J . Chem. Phys. 1975, 62, 1544; 1978, 68, 3891. (11) Heller, E. J. Acc. Chem. Res. 1981, 14, 368. (12) Reber, C.; Zink, J. 1. J. Phys. Chem. 1992. 96, 571. (13) Reber, C.; Zink, J. I. J . Chem. Phys. 1992, 96, 2681. (14) Alvarellos, J.; Metiu, H. J . Chem. Phys. 1988, 88, 4957. (15) Jiang, X.P.; Heather, R.; Metiu, H. J . Chem. Phys. 1989, 90, 2555. (16) Heather, R.; Metiu, H. J . Chem. Phys. 1989, 90, 6903. (17) Zhang, J.; Heller, E. J.; Huber, D.; Imre, D. G.; Tannor, D. J . Chem. Phys. 1988.89, 3602. (18) Das, S.; Tannor, D. J. Chem. Phys. 1989, 91, 2324. (19) Zhang, J.; Heller, E. J.; Huber, D.; Imre, D. G. J . Phys. Chem. 1991, 95, 6129. (20) Kosloff. D.; Kosloff, R. J . Comput. Phys. 1983, 52, 35. (21) For an introductory overview, see: Tanner, J. J. J . Chem. Educ. 1990, 67, 917.

J . Phys. Chem. 1992, 96, 8765-8777 (22) For simplicity, we choose harmonic excited-statepotentials in all of the following examples, although the theoretical method is not restricted by the functional form of the potentials. The potentials are given by V,(Q) = (1!2)k,(Q - AQ,)2 + E, with k! = 4r2.M(ho,)' being the force constant, AQ, being the position of the potential minimum along Q, and E, being the energy of the potential minimum for state i. These uncoupled potentials are shown as dashed lines in Figure 1 (diabatic potentials). The coupling between the diabatic potentials for states 1 and 2 is chosen to be coordinate independent; Le., V,,= V2, = constant. Again, the computational method allows us to use coordinate-dependent coupling. The most important coupling mechanism in transition-metal spectra is spin-orbit coupling, which does not strongly depend on nuclear coordinates. For simplicity, we assume a harmonic ground-state potential in all the examples presented here. The wave functions 4, at t = 0 are therefore Gaussians. Also for simplicity. the transition moments pi were chosen to be coordinate-independent, i.e., constants, in all of the following examples.

8765

(23) Hitchman, M. A.; Stratemeier, H.; Hoppe, R. Inorg. Chem. 1988,27, 2506. (24) Nowitzki, B.; Hoppe, R. Croat. Chem. Acta 1984, 57, 537. (25) Chang, T. H.; Zink, J. I. J . Am. Chem. SOC.1984, 106, 287. (26) Sugano, S.;Tanabe, Y.; Kamimura, H. Multiplets of Transition Metal Ions in Crystals; Academic Press: New York, 1970; Chapter 5.2. (27) Reber. C.; Zink, J. I. Comments Inorg. Chem. 1992, 13, 177. (28) Jacobsen, S.M.; Gudel, H. U.; Dad, C. A. J. Am. Chem. Soc. 1988, 110, 7610. (29) Greenough, P.; Paulusz, A. G. J . Chem. Phys. 1979, 70, 1967. (30) Wood,D. L.; Ferguson, J.; Knox, K.; Dillon, J. F. J. Chem. Phys. 1963, 39, 890. (31) Ferguson, J.; Guggenheim, H. J.; Wood, D. L. J . Chem. Phys. 1971, 54, 504. (32) Chodos, S.L.; Black, A. M.; Flint, C. D. J. Chem. Phys. 1976,65, 4816.

Stereodynamics of Sterically Crowded Metal-Phosphine Complexes: trans-[ (f-Bu),P(i-Pr)],MCi2 [M = Pt( I I ) and Pd( I I)]. One-Dimensional Dynamic and Two-Dlmenslonal Chemical Exchange NMR Studies, X-ray Crystallographic Studies, Molecular Conformation Trapping, and Molecular Mechanics Calculations Christine M. DiMeglio,*Vt Kazi J. Ahmed; Linda A. Luck,t Eugen E. Weltin,t Arnold L. Rheingold,t and C. Hackett BushweUer*st Departments of Chemistry, University of Vermont, Burlington, Vermont 05405, and University of Delaware, Newark, Delaware 1971 6 (Received: April 15, 1992; In Final Form: July 20, 1992)

By using complementary techniques including dynamic NMR (DNMR) spectroscopy, X-ray crystallography, molecular conformation trapping and molecular mechanics calculations, incisive pictures of the stereodynamics of two stericallycrowded Pd(I1) and Pt(I1) complexes have been elucidated. X-ray crystallography shows that the molecular geometry in the only crystalline modification (PZ,/a space group) of tr~ns-[(t-Bu)~P(i-Pr)]~PdCl~ (1) has C, symmetry. The isopropyl groups are mutually anti; the dihedral angle between the two P-CH bonds is 180'. For the isopropyl group on each phosphine ligand, one methyl group is anti and the other gauche to the palladium atom. There are significant torsions about Pd-P, P-(i-Pr) and P-(t-Bu) bonds including a dihedral angle between proximate P-CH and Pd-CI bonds of 21.6'. On the basis of the hypotheses that (a) when placed in solution, the conformation of each phosphine ligand of 1 will adopt an essentially exclusive preference for that in the crystal and (b) the two phosphines act essentially independently of each other, four stable diastereomericequilibrium conformationsare predicted. In two diasteromeric conformations, the isopropyl groups are mutually syn with dihedral angles between the P-CH bonds of 44' (C2symmetry) and ' 0 (C, symmetry). In the other two diastereomeric forms, the isopropyl groups are mutually anti with dihedral angles between the P-CH bonds of 136' (C2symmetry) and 1 80' (C, symmetry). Two-dimensional 31P('H)NMR chemical exchangespectroscopy in conjunction with theoretical simulations of the one-dimensional 31P{'HJand 13C{IH]DNMR spectra do reveal the presence of four diastereomeric equilibrium conformations and allow elucidation of the preferred conformational interconversion pathways. Dissolution of a sample of crystalline 1 at 150 K where conformational exchange is very slow on the laboratory time scale resulted in a solution of the pure C,-symmetric molecular conformation. The 31P(lH)NMR spectrum of this solution at 150 K and spectra recorded at higher temperatures, where equilibration to the other three stable conformationsoccurs, allow assignmentsof NMR resonanca to specific conformations and the calculation of relative conformational free energies at 200 K C, anti form (0.00 kcal/mol), C, syn (0.28), C, syn (0.29), C2anti (0.49). tr~ns-[(t-Bu)~P(i-Pr)]~PtCI, (2) exists in two crystalline modifications that have space groups P2,/a (C, anti molecular geometry) and P2,/c (C2anti molecular geometry). DNMR studies of 2 in solution reveal stereodynamicsthat are virtually identical to 1 with a similar distribution of relative conformational free energies at 210 K: C, anti (0.00 kcal/mol), C2syn (0.19), C, syn (OSl), C2anti (0.45). Molecular mechanics calculations performed for 1 by using a locally modified version of Allinger's 1985 MM2 force field agree qualitatively with the experimental results and provide important insight into the transition states for conformational exchange. For 1, the crystals belong to space group P2,/a with Z = 4, a = 15.057 (6) A, b = 12.318 (4) A, c = 16.242 (6) A, /3 = 112.88 (3)', V = 2775 (2) A3,R(F) = 4.1776, and R,(F) = 4.82%. For the P2,/a crystalline modification of 2, Z = 4, a = 15.096 (5) A, b = 12.307 (6) A, c = 16.255 (7) A, fi = 112.63 (3)', Y = 2787 (2) A', R(F) = 3.2476, and R,(F) = 3.91%. For the PZ,/ccrystalline modification of 2, Z = 4, a = 8.536 (2) A, b = 27.670 (6) A, c = 12.174 (3) A, fi = 107.11 (2)', V = 2748 (1) A3,R(F) = 3.75%, and R,(F) = 3.93%.

Introduction Tetracoordinate complexes of R(II), pd(II), lr(1), and Rh(I) that show square-planar coordination are ubiquitous in the ,-hemicai literature.12 These complexes constitute an important To whom correspondence should be addressed. + Universitv - ... . .. .., of - - Vermont. ---------

*University of Delaware.

0022-3654/92/2096-8765$03.00/0

class of chemical compounds. They have applications in catalysis and synthesis including asymmetric i n d ~ c t i o n . They ~ are inherently interesting from structural and conformational perspectives in that a plethora of ligands is available for complexation and, once formed, the complexes can exist as cis and trans geometrical isomers each of which may adopt a variety of equilibrium conformations. While myriad X-ray crystallographic studies have determined the preferred molecular conformation(s) in the 0 1992 American Chemical Society