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Cooperative optical transitions in impurity centers coupled via host atoms
Volume 138, number 2,3
CHEMICAL PHYSICS LETTERS
COOPERATIVE OPTICAL TRANSITIONS COUPLED VIA HOST ATOMS
17 July 1987
IN IMPURITY CENTERS
Isidore LAST ’ Departments of Chemistry and Physics &Astronomy, 239 Fronczak Hall, State University ofNew York at Buffalo, Buffalo, NY 14260, USA
Young Sik KIM ’ Department of Chemistry, University ofRochester, Rochester, NY 14627, USA
and Thomas F. GEORGE Departments of Chemistry and Physics &Astronomy, 239 Fronczak Hall, State University of New York at Buffalo, Buffalo, NY 14260, USA Received 17 March 1987; in final form 28 April 1987
In solids with a coupling between guest and host atoms, a new mechanism of cooperative transitions is possible since the centers formed by the guest atoms involve surrounding host atoms. This leads to orbital overlap between centers via the host atoms which can result in cooperative transitions. The cooperative transition moments are estimated for rare gas solids doped by halogens.
1. Introduction Cooperative optical transitions have been studied both experimentally [ l-51 and theoretically [6-l I] in crystals containing rare earth ions. In the simplest case, two atoms (ions) change their electronic state simultaneously, leading to absorption or emission of a photon with the energy approximately equal to the sum of the transition energies of separated atoms [ 1,4,6,7]. In more complicated cases the optical transitions include cooperative energy transfer between atoms [ 2,3,5,8,9,11] or two-photon absorption [ lo]. The cooperative optical transitions in the rare earth ions are connected with their inner electrons, and, consequently, the atomic orbitals involved do not overlap significantly with other atoms. The coupling which is responsible for the cooperative transitions is caused by a long-range dipole-dipole interaction [ 61. The same dipole-dipole interaction which causes the cooperative transitions in rare earth ions [6] is suggested to be responsible for the energy transfer between OH and NH guest molecules in a rare gas solid [ 121. Cooperative optical transitions have been detected in solid O2 and O,-Ar mixtures [ 13,141. In the case of molecules and non-rare-earth atoms, the transitions are connected with outer electrons so that the dipole-dipole interaction is not the only cause of cooperative transitions. The cooperative transitions in O2 molecules, for example, are attributed to direct overlap of molecular orbitals [ 151. We shall consider in the present paper a new mechanism of cooperative optical transitions which results from I Permanent address: Soreq Nuclear Research Center, Yavne 70600, Israel. ’ Present address: Departments of Chemistry and Physics &Astronomy, 239 Fronczak Hall, State University of New York at Buffalo, Buffalo, NY 14260, USA.
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indirect (via host atoms) overlap of the wavefunctions of different active centers formed by guest molecules (atoms). This mechanism is possible in solids where the molecular electronic states are coupled with the electronic states of host atoms. Due to this coupling, the molecular wavefunctions are partly delocalized as they involve surrounding host atoms. If there are host atoms which contribute non-negligible amplitudes to the wavefunctions of two guest molecules, then these wavefunctions overlap and the cooperative transition involving the two molecules can be realized. Systems with strongly delocalized Wannier-type excitons [ 16,171 will be excluded from this consideration. Generally speaking, the outer orbitals of guest molecules are always coupled to a certain degree with the orbitals of host atoms. However, this coupling can be very weak, like in rare gas solids doped by metallic atoms [ 18,191. Stronger coupling exists, most probably, in the case of low lying Rydberg excitations of guest atoms [ 201 and molecules [ 2 l-231. Relatively strong coupling is expected in rare gas solids doped by halogen atoms and molecules. Due to the positive electron affinity, the rare gas-halogen systems form excited ionic states with electron transfer to a halogen atom. The ionic rare gas halides, which are found in gas phase [ 241, solids [ 25-271 and liquids [28], have interesting optical, and in particular, lasing properties. In solids, the ionic states can include several matrix atoms forming a delocalized impurity center [ 291. Such delocalized centers can overlap one with another and consequently become capable of cooperative transitions. A weak delocalization takes place also in neutral states because of the mixing of neutral and ionic configurations [ 25,301. This mixing is expected to be stronger for halogen atoms than molecules [ 3 11. The weakness of the coupling between halogen molecules and the matrix is supported by experimental work which has found neither a noticeable perturbation of molecular valence states in solids nor intermolecular energy transfer [ 32,331. It follows that the most promising systems for the cooperative transitions are those which form ionic states. However, other systems, like rare gas solids with valence excited halogen molecules [ 32,331 and molecules or atoms in the lowest Rydberg states [ 20-231, must not be ruled out as candidates for cooperative transitions.
2. Cooperative transition moment Let us consider a rare gas solid matrix with a guest atom or molecule A whose electronic states are coupled with the electronic states of the host atoms forming an A center. Due to the coupling, the wavefunction of the center is the superposition of the wavefunctions of neutral configurations and ionic configurations with electron transfer from host atoms to the guest atom (molecule). It is more convenient to consider the ionic states as a hole transfer from the guest atom to the matrix and to describe the system by a hole wavefunction. Neglecting the perturbation of surrounding electrons by hole motion, we can use the one-electron (one-hole) approximation and express the A center wavefunction as a linear combination of atomic orbitals (AO) x by
where k is the state index, i= 1 is the index of the guest atom, i> 1 are the indexes of the host atoms, m stands for the A0 orientation (m= 1, 2 and 3 for p A0 and m is absent for the case of s AO), and ak,,,,, are the wavefunction coefficients. In the neutral states, the guest atom is heavily populated whereas the coefficients for the host atoms are small ( I a+, 1-Al, i> 1). In the ionic states, on the other hand, the guest atom coefficients are small ( 1ak.,,,,)el ) . We shall consider from now on a general case without distinguishing between the neutral and ionic states. In the zero overlap of atomic orbitals (ZOAO) approximation, the overlap integrals and the static and transition moments are expressed as follows:
(2)
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~Ak,k)=e(~,ak)(r-R,)~ak’)=e c W.,dR,-RI) riY”
=e(
v$,k’rvik’)
> =e
1
I
R,
1
m
ak,rmak.,rm
,
LE-lTERS
(3)
,
k’
zk
17 July 1987
(4)
,
where r is the electron radius vector, R, is the radius vector of the ith atom, and w,,ki is the population of the ith atom WA.kr
=
1 (ak,rm)* WI
(5)
-
The cooperative two-electron transition involving two centers (A and B) is described by the two-electron transition moment (6)
where YL”, describes the centers A and B in the states k and I, respectively. Using the Lijwdin orthogonalization [ 341, we can present the singlet state function !P$,k, orthogonalized to !Pj,$“) in the following form:
~ak,==2-“*[1-t(S~~‘)*l[~vak’(r,)
wV’(~*)+WP(~*)
-2-“*Sakgl’)S~~)[Wak’)(r,) ty&“)(r,)-wv’)(r,)
wLk’(r2)l
,
yi”‘)(r2)]
(7)
where Sj,%) are the intercenter overlap integrals S%? = (WA”’WB”> = 1 1 I
~k,dh
(8)
,
m
are the B center wavefunction coefficients (see eq. (1)) . The overlap integrals (8) are assumed to be h.,,,, small, which was taken into account in eqs. (7) by neglecting the terms of the order higher than (Sj$)*. Substituting the wavefunctions !P@ and !Pj,%P)into the expression in eq. (6) and neglecting the higherorder terms, one obtains the transition moment for the simultaneous (cooperative) transitions k-tk’ in the center A and l+l’ in the center B, and
,,k/__k,r=Sj,~“[~~$,’ - &~‘(,@’
+,#$‘)]
+Sj,~‘[,@,)
- j&#$,‘(,,& +p$,kbk”)] ,
(9)
where pAAand pBa are the static dipole moments (see eq. (3)) and ~*a are one-electron transition moments between the centers A and B, (10)
The transition moments in eq. (10) are expected to be small, on the order of the overlap integrals of eq. (8). The cooperative transition moment is on the order of the square of the overlap integrals and is non-zero only for the case where there are matrix atoms which are populated by both centers.
3. Numerical estimations of the cooperative transition moments Cooperative transitions can be detected if the two-electron (cooperative) transition moment is not very small. We shall perform the cooperative transition moment estimations for a model system consisting of five atoms and in a Xe solid doped by Cl atoms. Here, each of the active centers (A and B) consists of three S-symmetry atoms of collinear geometry. One of the atoms is common for both centers so that the total number of atoms is 5 (fig. 1). The atoms in the middle of the active centers (i = 2 and 4) are considered as guest atoms, whereas 227
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Fig. 1. The arrangement of atoms forming the active centers A (1,2and3atoms)andB(3,4and5atoms).
the other three atoms (i= 1, 3 and 5) are considered as host atoms. The centers are assumed to be unperturbed one by another, as well as by surrounding matrix atoms. The guest atoms are shifted towards the margin atoms i= 1 and 5, so that R,
+V&+2
(11)
9
w(k’)=[-1+(1-2-“2)r12+ly2]X,+Y[1+(21’2-l)tl2]X,+1-2-1’2~~,+2,
(11’)
where j= 1 and 3 stand for the A and B centers, respectively, and the coefficients y and q are small, i.e. y’s1 and n2=zzzl.The initial and final states of both centers are assumed to be the same, i.e. I= k and I’ = k' . Sub-
stituting the coefficients of the wavefunctions in eqs. (11) into eqs. (8)-( lo), one obtains the cooperative transition moment (in D) /‘kk_k.k,=2.401q4y2(R, +2R2)
COS(i8)
,
(12)
where R are in A. The cooperative transition moment in eq. (12) depends sharply on the parameter q which determines the amplitude of the common atoms (i= 3) in the wavefunctions in eq. (11). The coefficients y and r7were obtained in the semi-empirical calculations of the Xe-Cl-Xe system [ 3 11. For the distance RI = 3.2 A, which is close to the van der Waals Xe-Cl distance, and R2 = 3.6 A, the semi-empirical calculation gives y = 0.16 and q= 0.57. Substituting these values in eq. (12) for 19= 90”) one obtains a reasonably sized transition moment of 0.048 D. The increase of the R2 distance to R2=4 8, decreases the coefficient of,which becomes equal to 0.32. Consequently, the cooperative transition moment becomes very small, 0.005 D only. In the case of the Xe solid, two guest Cl atoms are located in adjacent cages whose radius is 4.33 A. Each of the Cl atoms forms together with the cage Xe atoms an active center which can be excited to an ionic state. The coefficients of the wavefunction describing an active center have been obtained in a semi-empirical calculation [ 31,351. The coefficients depend on the location of the Cl atom, which can be moving inside a Xe cage in the limits of the radius RG 1 A at the temperature Tx 10 K [ 351. The calculation of the transition moment in eq. (9) was performed for different locations of the Cl atoms inside a fixed Xe cage, excluding only the case when both Cl atoms are located close to one Xe atom and the coupling between the centers is strong. Some of the results of the calculation are presented in fig. 2, where the cooperative transition moments are given in the common log scale as functions of the Cl atom shift R from the cage center. The cooperative transition moment reaches relatively large values, on the order of 0.1-0.4 D, when both Cl atoms are shifted simultaneously to one host atom (curve (a) of fig. 2). In another case (curve (b) of fig. 2), when one of the two Cl atoms remains in the cage center whereas the second atom is shifted from its center along the line connecting both Cl atoms, the cooperative transition moment does not exceed 0.006 D. The values obtained for the cooperative transition moments are not negligible small, and in some cases (curve (a) of fig. 2) even relatively large. The result promises the possibility of the experimental detection of the cooperative absorption in the Xe solid containing Cl atoms. The cooperative absorption is expected in the region about 150 nm as the usual one-
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Fig. 2. Common log of the cooperative transition moments for two guest Cl atoms located in adjacent Xe cages. (a) Both Cl atoms are shifted from their cage centers at the distance R to the direction of one common Xe atom. (b) One of the Cl atoms is fixed at its cage center, where-asanother Cl atom is shifted towards the first one by the distance R.
center absorption lies in the region about 300 nm [ 271. The probability of finding two Cl atoms in adjacent cages may not be low even for low Cl concentration, since the pair of Cl atoms is formed by the photodissociation of one Cl2 molecule. The calculations performed above have to be considered as relatively rough estimations which give the order of the magnitude of the cooperative transition moments. In order to get results of higher accuracy, one needs a more sophisticated many-electron theory of the phenomenon.
Acknowledgement
This research was supported by the Air Force Office of Scientific Research (AFSC), United State Air Force, under Contract F49620-86-C-0009, the Office of Naval Research and the National Science Foundation under Grant CHE-85 19053. We thank Professor V.A. Apkarian for helpful discussions of various aspects of the problem. IL thanks SUNY-Buffalo for its hospitality during a visit in 1986-87 when this work was performed.
References [ 1 ] F. Varsanyi and G.H. Dieke, Phys. Rev. Letters 7 ( 196 1) 442. [ 21 V.V. Ovsyankin and P.P. Feofilov, Soviet Phys. JETP Letters 4 (1966) 317. [ 3) R.A. Hewes and J.W. Sarver, Phys. Rev. 182 (1969) 427. [4] E. Nakazawa and S. Shionoya, Phys. Rev. Letters 25 (1970) 17 IO. [ 51F.E. Auzel, Proc. IEEE 61 (1973) 758. [ 61 D.L. Dexter, Phys. Rev. 126 ( 1962) 1962. [ 71 A.V. Tulub and K. Patzer, Phys. Stat. Sol. 26 (1968) 693. [ 81 T. Miyakawa and D.L. Dexter, Phys. Rev. Bl (1970) 70. [9] T. Kushida, J. Phys. Sot. Japan 34 (1973) 1318. [lo] J.R.R. Leiteand C.B. de Araujo, Chem. Phys. Letters 73 (1980) 71. [ll]J.F.LamandS.C.Rand,Phys.Rev.A35 (1987)2164. [ 121 L.E. Brus and V.E. Bondybey, J. Chem. Phys. 63 (1975) 786; V.E. Bondybey and L.E. Brus, J. Chem. Phys. 63 (1975) 794. [ 131 D.D. Awschalom, K. Iwahana, D.B. Fitchen and S. Gregory, Phys. Rev. B25 (1982) 4185. [ 141 I.R. Sil’dos, L.A. Rebane, A.B. Treshchalov and A.E. Lykhmus, Soviet JETP Letters 6 (1975) 15 I. [ 151 R. Bhandari and L.M. Falicov, J. Phys. C6 (1973) 479. [ 161 B. Raz and J. Jortner, Proc. Roy. Sot. A3 17 ( 1970) 113. [ 171 N. Schwenter, E.E. Koch and J. Jortner, Electronic excitations in condensed rare gases (Springer, Berlin, 1985).
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[ 181 D.M. Kolb and D. Leutloff, Chem. Phys. Letters 55 (1978) 264. [ 191 F. Forstmann and S. Ossicini, J. Chem. Phys. 73 (1980) 5997; S. Ossicini and F. Forstmann, J. Chem. Phys. 75 (198 1) 2076. [20] S. Webber, S.A. Rice and J. Jortner, J. Chem. Phys. 42 (1965) 1907. [21] J. Goodmann and L.E. Brus, J. Chem. Phys. 67 (1977) 933. (221 M. Chergui, N. Schwentner and W. Biihmer, J. Chem. Phys. 85 (1986) 2472. [23] E. Boursey, J. Chem. Phys. 62 (1975) 3353. [24] S.K. Searles and G.A. Hart, Appl. Phys. Letters 27 (1975) 243; C.A. Brau and J.J. Ewing, J. Chem. Phys. 63 (1975) 4640; V.S. Dubov, L.I. Gudzenko, L.V. Gurvich and S.I. Iakovlenko, Chem. Phys. Letters 53 (1976) 170; A.W. McCown and J.C. Eden, J. Chem. Phys. 8 I (1984) 2933. [25] B.S. Ault and L. Andrews, J. Chem. Phys. 65 (1976) 4192. [ 261 J. le Calve and M. Chergui, Chem. Phys. Letters 132 (1986) 256. [27] M.E. Fajardoand V.A. Apkarian, J. Chem. Phys. 85 (1986) 5660. [28] L. Wiedeman, M.E. Fajardo and V.A. Apkarian, submitted for publication. [29] I. Last and T.F. George, J. Chem. Phys. 86 (1987) 3787. [30] P.J. Hay and T.H. Dunning Jr., J. Chem. Phys. 69 (1978) 2209. [ 3 1] I. Last and T.F. George, J. Chem. Phys., submitted for publication. [ 321 V.E. Bondybey and C. Fletcher, J. Chem. Phys. 64 (1976) 36 15. [ 331 M. Mandlich, P. Beeken and G. Flynn, J. Chem. Phys. 77 (1982) 702. [ 341 EL. Pilar, Elementary quantum chemistry (McGraw-Hill, New York, 1968) p. 565. [ 351 I. Last, T.F. George, M.E. Fajardo and V.A. Apkarian, J. Chem. Phys., submitted for publication.
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CHEMICAL PHYSICS LETTERS
COOPERATIVE OPTICAL TRANSITIONS COUPLED VIA HOST ATOMS
17 July 1987
IN IMPURITY CENTERS
Isidore LAST ’ Departments of Chemistry and Physics &Astronomy, 239 Fronczak Hall, State University ofNew York at Buffalo, Buffalo, NY 14260, USA
Young Sik KIM ’ Department of Chemistry, University ofRochester, Rochester, NY 14627, USA
and Thomas F. GEORGE Departments of Chemistry and Physics &Astronomy, 239 Fronczak Hall, State University of New York at Buffalo, Buffalo, NY 14260, USA Received 17 March 1987; in final form 28 April 1987
In solids with a coupling between guest and host atoms, a new mechanism of cooperative transitions is possible since the centers formed by the guest atoms involve surrounding host atoms. This leads to orbital overlap between centers via the host atoms which can result in cooperative transitions. The cooperative transition moments are estimated for rare gas solids doped by halogens.
1. Introduction Cooperative optical transitions have been studied both experimentally [ l-51 and theoretically [6-l I] in crystals containing rare earth ions. In the simplest case, two atoms (ions) change their electronic state simultaneously, leading to absorption or emission of a photon with the energy approximately equal to the sum of the transition energies of separated atoms [ 1,4,6,7]. In more complicated cases the optical transitions include cooperative energy transfer between atoms [ 2,3,5,8,9,11] or two-photon absorption [ lo]. The cooperative optical transitions in the rare earth ions are connected with their inner electrons, and, consequently, the atomic orbitals involved do not overlap significantly with other atoms. The coupling which is responsible for the cooperative transitions is caused by a long-range dipole-dipole interaction [ 61. The same dipole-dipole interaction which causes the cooperative transitions in rare earth ions [6] is suggested to be responsible for the energy transfer between OH and NH guest molecules in a rare gas solid [ 121. Cooperative optical transitions have been detected in solid O2 and O,-Ar mixtures [ 13,141. In the case of molecules and non-rare-earth atoms, the transitions are connected with outer electrons so that the dipole-dipole interaction is not the only cause of cooperative transitions. The cooperative transitions in O2 molecules, for example, are attributed to direct overlap of molecular orbitals [ 151. We shall consider in the present paper a new mechanism of cooperative optical transitions which results from I Permanent address: Soreq Nuclear Research Center, Yavne 70600, Israel. ’ Present address: Departments of Chemistry and Physics &Astronomy, 239 Fronczak Hall, State University of New York at Buffalo, Buffalo, NY 14260, USA.
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indirect (via host atoms) overlap of the wavefunctions of different active centers formed by guest molecules (atoms). This mechanism is possible in solids where the molecular electronic states are coupled with the electronic states of host atoms. Due to this coupling, the molecular wavefunctions are partly delocalized as they involve surrounding host atoms. If there are host atoms which contribute non-negligible amplitudes to the wavefunctions of two guest molecules, then these wavefunctions overlap and the cooperative transition involving the two molecules can be realized. Systems with strongly delocalized Wannier-type excitons [ 16,171 will be excluded from this consideration. Generally speaking, the outer orbitals of guest molecules are always coupled to a certain degree with the orbitals of host atoms. However, this coupling can be very weak, like in rare gas solids doped by metallic atoms [ 18,191. Stronger coupling exists, most probably, in the case of low lying Rydberg excitations of guest atoms [ 201 and molecules [ 2 l-231. Relatively strong coupling is expected in rare gas solids doped by halogen atoms and molecules. Due to the positive electron affinity, the rare gas-halogen systems form excited ionic states with electron transfer to a halogen atom. The ionic rare gas halides, which are found in gas phase [ 241, solids [ 25-271 and liquids [28], have interesting optical, and in particular, lasing properties. In solids, the ionic states can include several matrix atoms forming a delocalized impurity center [ 291. Such delocalized centers can overlap one with another and consequently become capable of cooperative transitions. A weak delocalization takes place also in neutral states because of the mixing of neutral and ionic configurations [ 25,301. This mixing is expected to be stronger for halogen atoms than molecules [ 3 11. The weakness of the coupling between halogen molecules and the matrix is supported by experimental work which has found neither a noticeable perturbation of molecular valence states in solids nor intermolecular energy transfer [ 32,331. It follows that the most promising systems for the cooperative transitions are those which form ionic states. However, other systems, like rare gas solids with valence excited halogen molecules [ 32,331 and molecules or atoms in the lowest Rydberg states [ 20-231, must not be ruled out as candidates for cooperative transitions.
2. Cooperative transition moment Let us consider a rare gas solid matrix with a guest atom or molecule A whose electronic states are coupled with the electronic states of the host atoms forming an A center. Due to the coupling, the wavefunction of the center is the superposition of the wavefunctions of neutral configurations and ionic configurations with electron transfer from host atoms to the guest atom (molecule). It is more convenient to consider the ionic states as a hole transfer from the guest atom to the matrix and to describe the system by a hole wavefunction. Neglecting the perturbation of surrounding electrons by hole motion, we can use the one-electron (one-hole) approximation and express the A center wavefunction as a linear combination of atomic orbitals (AO) x by
where k is the state index, i= 1 is the index of the guest atom, i> 1 are the indexes of the host atoms, m stands for the A0 orientation (m= 1, 2 and 3 for p A0 and m is absent for the case of s AO), and ak,,,,, are the wavefunction coefficients. In the neutral states, the guest atom is heavily populated whereas the coefficients for the host atoms are small ( I a+, 1-Al, i> 1). In the ionic states, on the other hand, the guest atom coefficients are small ( 1ak.,,,,)el ) . We shall consider from now on a general case without distinguishing between the neutral and ionic states. In the zero overlap of atomic orbitals (ZOAO) approximation, the overlap integrals and the static and transition moments are expressed as follows:
(2)
226
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2,3
PHYSICS
~Ak,k)=e(~,ak)(r-R,)~ak’)=e c W.,dR,-RI) riY”
=e(
v$,k’rvik’)
> =e
1
I
R,
1
m
ak,rmak.,rm
,
LE-lTERS
(3)
,
k’
zk
17 July 1987
(4)
,
where r is the electron radius vector, R, is the radius vector of the ith atom, and w,,ki is the population of the ith atom WA.kr
=
1 (ak,rm)* WI
(5)
-
The cooperative two-electron transition involving two centers (A and B) is described by the two-electron transition moment (6)
where YL”, describes the centers A and B in the states k and I, respectively. Using the Lijwdin orthogonalization [ 341, we can present the singlet state function !P$,k, orthogonalized to !Pj,$“) in the following form:
~ak,==2-“*[1-t(S~~‘)*l[~vak’(r,)
wV’(~*)+WP(~*)
-2-“*Sakgl’)S~~)[Wak’)(r,) ty&“)(r,)-wv’)(r,)
wLk’(r2)l
,
yi”‘)(r2)]
(7)
where Sj,%) are the intercenter overlap integrals S%? = (WA”’WB”> = 1 1 I
~k,dh
(8)
,
m
are the B center wavefunction coefficients (see eq. (1)) . The overlap integrals (8) are assumed to be h.,,,, small, which was taken into account in eqs. (7) by neglecting the terms of the order higher than (Sj$)*. Substituting the wavefunctions !P@ and !Pj,%P)into the expression in eq. (6) and neglecting the higherorder terms, one obtains the transition moment for the simultaneous (cooperative) transitions k-tk’ in the center A and l+l’ in the center B, and
,,k/__k,r=Sj,~“[~~$,’ - &~‘(,@’
+,#$‘)]
+Sj,~‘[,@,)
- j&#$,‘(,,& +p$,kbk”)] ,
(9)
where pAAand pBa are the static dipole moments (see eq. (3)) and ~*a are one-electron transition moments between the centers A and B, (10)
The transition moments in eq. (10) are expected to be small, on the order of the overlap integrals of eq. (8). The cooperative transition moment is on the order of the square of the overlap integrals and is non-zero only for the case where there are matrix atoms which are populated by both centers.
3. Numerical estimations of the cooperative transition moments Cooperative transitions can be detected if the two-electron (cooperative) transition moment is not very small. We shall perform the cooperative transition moment estimations for a model system consisting of five atoms and in a Xe solid doped by Cl atoms. Here, each of the active centers (A and B) consists of three S-symmetry atoms of collinear geometry. One of the atoms is common for both centers so that the total number of atoms is 5 (fig. 1). The atoms in the middle of the active centers (i = 2 and 4) are considered as guest atoms, whereas 227
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Fig. 1. The arrangement of atoms forming the active centers A (1,2and3atoms)andB(3,4and5atoms).
the other three atoms (i= 1, 3 and 5) are considered as host atoms. The centers are assumed to be unperturbed one by another, as well as by surrounding matrix atoms. The guest atoms are shifted towards the margin atoms i= 1 and 5, so that R,
+V&+2
(11)
9
w(k’)=[-1+(1-2-“2)r12+ly2]X,+Y[1+(21’2-l)tl2]X,+1-2-1’2~~,+2,
(11’)
where j= 1 and 3 stand for the A and B centers, respectively, and the coefficients y and q are small, i.e. y’s1 and n2=zzzl.The initial and final states of both centers are assumed to be the same, i.e. I= k and I’ = k' . Sub-
stituting the coefficients of the wavefunctions in eqs. (11) into eqs. (8)-( lo), one obtains the cooperative transition moment (in D) /‘kk_k.k,=2.401q4y2(R, +2R2)
COS(i8)
,
(12)
where R are in A. The cooperative transition moment in eq. (12) depends sharply on the parameter q which determines the amplitude of the common atoms (i= 3) in the wavefunctions in eq. (11). The coefficients y and r7were obtained in the semi-empirical calculations of the Xe-Cl-Xe system [ 3 11. For the distance RI = 3.2 A, which is close to the van der Waals Xe-Cl distance, and R2 = 3.6 A, the semi-empirical calculation gives y = 0.16 and q= 0.57. Substituting these values in eq. (12) for 19= 90”) one obtains a reasonably sized transition moment of 0.048 D. The increase of the R2 distance to R2=4 8, decreases the coefficient of,which becomes equal to 0.32. Consequently, the cooperative transition moment becomes very small, 0.005 D only. In the case of the Xe solid, two guest Cl atoms are located in adjacent cages whose radius is 4.33 A. Each of the Cl atoms forms together with the cage Xe atoms an active center which can be excited to an ionic state. The coefficients of the wavefunction describing an active center have been obtained in a semi-empirical calculation [ 31,351. The coefficients depend on the location of the Cl atom, which can be moving inside a Xe cage in the limits of the radius RG 1 A at the temperature Tx 10 K [ 351. The calculation of the transition moment in eq. (9) was performed for different locations of the Cl atoms inside a fixed Xe cage, excluding only the case when both Cl atoms are located close to one Xe atom and the coupling between the centers is strong. Some of the results of the calculation are presented in fig. 2, where the cooperative transition moments are given in the common log scale as functions of the Cl atom shift R from the cage center. The cooperative transition moment reaches relatively large values, on the order of 0.1-0.4 D, when both Cl atoms are shifted simultaneously to one host atom (curve (a) of fig. 2). In another case (curve (b) of fig. 2), when one of the two Cl atoms remains in the cage center whereas the second atom is shifted from its center along the line connecting both Cl atoms, the cooperative transition moment does not exceed 0.006 D. The values obtained for the cooperative transition moments are not negligible small, and in some cases (curve (a) of fig. 2) even relatively large. The result promises the possibility of the experimental detection of the cooperative absorption in the Xe solid containing Cl atoms. The cooperative absorption is expected in the region about 150 nm as the usual one-
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Fig. 2. Common log of the cooperative transition moments for two guest Cl atoms located in adjacent Xe cages. (a) Both Cl atoms are shifted from their cage centers at the distance R to the direction of one common Xe atom. (b) One of the Cl atoms is fixed at its cage center, where-asanother Cl atom is shifted towards the first one by the distance R.
center absorption lies in the region about 300 nm [ 271. The probability of finding two Cl atoms in adjacent cages may not be low even for low Cl concentration, since the pair of Cl atoms is formed by the photodissociation of one Cl2 molecule. The calculations performed above have to be considered as relatively rough estimations which give the order of the magnitude of the cooperative transition moments. In order to get results of higher accuracy, one needs a more sophisticated many-electron theory of the phenomenon.
Acknowledgement
This research was supported by the Air Force Office of Scientific Research (AFSC), United State Air Force, under Contract F49620-86-C-0009, the Office of Naval Research and the National Science Foundation under Grant CHE-85 19053. We thank Professor V.A. Apkarian for helpful discussions of various aspects of the problem. IL thanks SUNY-Buffalo for its hospitality during a visit in 1986-87 when this work was performed.
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