Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 151 (2015) 510–514
Contents lists available at ScienceDirect
Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy journal homepage: www.elsevier.com/locate/saa
Effect of curve crossing induced dissociation on absorption and resonance Raman spectra: An analytically solvable model Diwaker ⇑, Aniruddha Chakraborty Indian Institute of Technology Mandi, Mandi (H.P.) 175001, India
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
Model consisting of harmonic
oscillator and Morse potential is solved. Dirac Delta function and Greens function is used. Absorption spectra and resonance Raman excitation profile is studied.
a r t i c l e
i n f o
Article history: Received 17 April 2015 Received in revised form 27 June 2015 Accepted 1 July 2015 Available online 3 July 2015
a b s t r a c t An analytically solvable model for the crossing of a harmonic and a Morse potential coupled by Dirac Delta function has been proposed. Further we explore the electronic absorption spectra and resonance Raman excitation profile using this model and found that curve crossing had significant effect on the resonance Raman excitation profile. Ó 2015 Elsevier B.V. All rights reserved.
Keywords: Quantum mechanics Scattering Multi-channel Analytical model Greens function
1. Introduction In various atomic and molecular processes we have different type of transitions which occur between two atomic/molecular states and it is worthwhile to say that potential energy curve crossing plays a key role in such kind of transitions. Nonadiabatic transitions (transitions between two adiabatic states or diabatic states) generally occur between bound states [1]. Nonadiabatic transitions also play a very important role in change of state/phase [2–6] in different kind of dynamic processes occurring in the field of ⇑ Corresponding author. E-mail address: [email protected] ( Diwaker). http://dx.doi.org/10.1016/j.saa.2015.07.004 1386-1425/Ó 2015 Elsevier B.V. All rights reserved.
biology, chemistry and physics. Similarly, various spectroscopic, collisions processes and different kind of reactions are governed by nonadiabatic transitions at crossing or avoided crossing of the potential energy surfaces [7]. Radiationless transitions in condense matter physics, flouroscence quenching, self- trapping of excitons, laser assisted collisions reactions are some of the other examples [8,9] where nonadiabatic transitions due to curve crossing or avoided crossing play an important role. Other important fields of study related to nonadiabatic transitions include the Zener transitions in flux driven matellic rings [10], super conducting Josephson junctions [11], nuclear collisions and reactions in nuclear physics [12], electron proton transfer processes in biological molecules [13]. Such kind of transitions also play a very
511
Diwaker, A. Chakraborty / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 151 (2015) 510–514
important role in neutrino conversion in the sun and provide an evidence for its existence in the mutability of the universe [14]. It has been also commented by Shaik [15] that curve crossing/ avoided crossing could be interpreted as a general mechanism of rearrangement of electrons by molecules leading to the formation and breaking of bonds. Nonadiabatic transitions have been also studied as a mechanism of dissociation of molecules on metal surfaces as reported by Kosloff [16] in his studies. The first report on such kind of transitions came into existence into 1932, when the pioneering works were published by Landau [17], Zener [18] and Stueckelberg [19] and by Rosen and Zener [20]. Since1932, there are numerous research papers have been published in this area using analytical as well as computational approach [21–33]. In our earlier publications we have reported analytical solution in those cases where two or more arbitrary potentials are coupled by Dirac Delta interactions [34,37,35,38,36,39]. Recently we have extended our research to deal with the cases where two potentials are coupled by any arbitrary interaction [40]. One of the great advantage of Dirac Delta coupling model is that we can find exact analytical solution using this model.
H¼
H1 ðxÞ VðxÞ ; VðxÞ H2 ðxÞ
where Hi ðxÞ is
Hi ðxÞ ¼
1 @2 þ V i ðxÞ: 2m @x2
UðxÞ ¼
Z
1
ð1Þ
In the above equation x is the nuclear coordinate and xc is considered as crossing point. V 1 ðxÞ and V 2 ðxÞ are the diabatic potentials (and Vðxc ) represent the coupling between them. In real systems the transitions between the two diabatic potentials occurs at the crossing point. This can be attributed to the fact that necessary energy transfer between the electronic and nuclear degrees of freedom is minimum at this point, hence it is worthful to analyze a coupling which is localized near the crossing point rather than using a model where coupling is constant or same elsewhere. Thus we put (see Appendix A for more details on diabatic coupling, where we do a calculation on real system like Lithium chloride i.e. (Eq. (2), Fig. 4))
VðxÞ ¼ k0 dðx xc Þ;
3. Exact analytical approach In this section we derive exact analytical expressions for wave function and Green’s function which are needed to study the effect of curve crossing on electronic absorption and resonance Raman excitation profile. We write the probability amplitude for the two included states as
/1 ðx; tÞ /2 ðx; tÞ
;
ð3Þ
where /1 ðx; tÞ and /2 ðx; tÞ are the probability amplitude for the two states. Uðx; tÞ obey the time dependent Schrodinger equation. In the h ¼ 1) subsequent calculations as well as the present one (we take
i
@ Uðx; tÞ ¼ HUðx; tÞ: @t H is given by
ð4Þ
ð7Þ
writing the half Fourier transform of Eq. (4) gave us
UðxÞ ¼ iGðxÞUð0Þ;
ð8Þ
where GðxÞ is defined by
ðx HÞGðxÞ ¼ I:
ð9Þ
In the position representation, the above equation may be written as
Z
1
Gðx; x0 ; xÞUðx0 ; xÞdx0 ;
ð10Þ
1
where Gðx; x0 ; xÞ is
Gðx; x0 ; xÞ ¼ hxjðx HÞ1 jx0 i:
ð11Þ
Writing
Gðx; x0 ; xÞ ¼
G11 ðx; x0 ; xÞ G12 ðx; x0 ; xÞ
ð12Þ
G21 ðx; x0 ; xÞ G22 ðx; x0 ; xÞ
and applying the partitioning technique we can write 1
G11 ðx; x0 ; xÞ ¼ hxj½x H1 Vðx H2 Þ1 V jx0 i:
ð13Þ
The above equation is true for any general potential VðxÞ. Our equation is further simplified if V is a delta function located at the crossing point
G11 ðx; x0 ; xÞ ¼ G01 ðx; x0 ; xÞ þ
ð2Þ
here k0 is a constant. This model has the advantage that it can be analytically solved.
Uðx; tÞ ¼
UðtÞeixt dt:
0
Uðx; xÞ ¼ i
jV 1 ðxÞ V 2 ðxÞj ’ jVðxc Þj:
ð6Þ
the half Fourier transform UðxÞ of UðtÞ is defined by
2. Mathematical model We consider the crossing of two diabatic curves (a harmonic potential and a Morse potential in the present case) and the coupling between them is assumed to be a Dirac Delta function which is responsible for transitions between these two considered diabatic curves. Nonadiabatic transition due to curve crossing would occurs in the neighborhood of the crossing point or in other words is given by
ð5Þ
K 20 G01 ðx; xc ; xÞG02 ðxc ; xc ; xÞG01 ðxc ; x0 ; xÞ 1 K 20 G01 ðxc ; xc ; xÞG02 ðxc ; xc ; xÞ
;
ð14Þ
where
G0i ðx; x0 ; xÞ ¼ hxjðx Hi Þ1 jx0 i;
ð15Þ
The above value of Green’s function corresponds to propagation of the particle starting at x0 on the second diabatic curve, in the absence of coupling to the first diabatic curve. In a similar fashion one can get
G12 ðx; x0 ; xÞ ¼
K 0 G01 ðx; xc ; xÞG02 ðxc ; x0 ; xÞ 1 K 20 G01 ðxc ; xc ; xÞG02 ðxc ; xc ; xÞ
:
ð16Þ
The expressions for G22 ðx; x0 ; xÞ and G21 ðx; x0 ; xÞ can be further derived by using a similar approach as shown in the last section. We can calculate UðxÞ explicitly by using these expressions for the Green’s function in Eq. (8). The expressions that we have obtained for UðxÞ are quite general and are valid for any V 1 ðxÞ and V 2 ðxÞ. However, their utility is limited by the fact that one must know G01 ðx; x0 ; xÞ and G02 ðx; x0 ; xÞ. It is possible to find G0i ðx; x0 ; xÞ only in a few limited cases and the Morse oscillator is one of them [41].
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4. Curve crossing induced dissociation: electronic absorption and resonance Raman excitation profile In the present section the mathematical model along with necessary derivations derived above have been used to solve the problem involving a harmonic oscillator and a Morse oscillator, coupled by Dirac Delta function. A system involving three Potential energy curves has been considered out of which we have a ground electronic state and two ‘crossing’ excited electronic states. Electronic transition to one of them is assumed to be dipole forbidden and while it is allowed to the other. We calculate the effect of ‘crossing’ induced dissociation on electronic absorption spectra and on resonance Raman excitation profile. The propagating wave functions on the excited state potential energy curves are given by solution of the time dependent Schrödinger equation
@ i @t
U1v ib ðx; tÞ
!
v ib
U2 ðx; tÞ
¼
Hv ib;e1 ðxÞ
V 12 ðxÞ
V 21 ðxÞ
Hv ib;e2 ðxÞ
U1v ib ðx; tÞ v ib
U2 ðx; tÞ
! :
ð17Þ
In the above equation Hv ib;e1 ðxÞ and Hv ib;e2 ðxÞ describes the vibrational motion of the system in the first electronic excited state which is allowed and second electronic excited state which is forbidden
Hv ib;e1 ðxÞ ¼
1 @2 1 þ mx2A ðx aÞ2 2m @x2 2
ð18Þ
1 @2 2 þ DF ½1 eðxbÞ : 2m @x2
ð19Þ
and
Hv ib;e2 ðxÞ ¼
In the above m is the oscillator’s mass, xA is the vibrational frequency of the first electronic excited state and DF is the dissociation energies of the forbidden states and x is the vibrational coordinate. Shifts of the vibrational coordinate minimum upon excitation are given by a and b, and V 12 ðxÞ (V 21 ðxÞ) represent coupling between the two diabatic potentials which is taken to be
V 21 ðxÞ ¼ V 12 ðxÞ ¼ K 0 dðx xc Þ;
ð20Þ
where K 0 represent the strength of the coupling. The equation representing the intensity of electronic absorption spectra can be written as
Z IA ðxÞ / Re
dx0 Uvi ib ðxÞiGðx; x0 ; x þ iCÞUiv ib ðx0 Þ ;
ð21Þ
E D 1 Gðx; x0 ; x þ iCÞ ¼ x½ðx0 =2 þ x xeg Þ þ iC Hv ib;e x0 :
ð22Þ
1
dx
Z
1
1
1
way we can write the equation representing resonance Raman scattering intensity in terms of Green’s function and is given by.
Z IR ðxÞ /
1
dx
1
Z
1 1
2 dx0 Ufv ib ðx; 0ÞiGðx; x0 ; x þ iCÞUiv ib ðx0 ; 0Þ : ð25Þ
In the above Ufv ib ðx; 0Þ is given by
Ufv ib ðx; 0Þ ¼
vf ðxÞ 0
;
ð26Þ
vf ðxÞ is the final vibrational state of the ground electronic state. As G0i ðx; x0 ; xÞ for the harmonic potential is known, we can calculate Gðx; x0 ; xÞ. We use Eq. (25) to calculate the effect of curve where
crossing induced dissociation on resonance Raman excitation profile.
5. Results In the following section we discussed the results of curve crossing on both the electronic absorption spectra and resonance Raman excitation profile as shown in Fig. 1. The ground state curve is considered to be a harmonic curve with its minimum at zero. This curve can be exactly mapped to the case of potential energy curve produced in stretching of a metal ligand bond. We took mass as 35 a.m.u. in our calculations and 400 cm1 as the vibrational wavenumber for the ground state. The first diabatic excited state to which transitions are allowed is displaced by 0.1 Å. Its minimum is taken to the above 10,700 cm1 in comparison to ground state curve. The same parameters for another diabatic curve to which transitions are forbidden are taken as 400 cm1 and 10,800 cm1 respectively. These two excited diabatic curves crosses at an energy of 10; 804:1 cm1 . Value of xc and k0 in our calculations are 0:02477 Å and 5:54277 1015 erg Å respectively. 450 cm1 is taken to be as life time of both the excited states. The result of our calculations are shown in Figs. 2 and 3 respectively. The profile shown by the dashed line is in the absence of any coupling to the second potential energy curve while the profile shown in solid line has the effect of coupling in it. It is seen that curve crossing effect can alter the absorption and Raman excitation
where
and
Hv ib;e ¼
Hv ib;e1 ðxÞ
K 0 jxc ihxc j
K 0 jxc ihxc j
Hv ib;e2 ðxÞ
ð23Þ
Here, C is a phenomenological damping constant which account for the life time effects. Uiv ib ðx; 0Þ is given by
Uvi ib ðx; 0Þ ¼
vi ðxÞ 0
;
ð24Þ
where the symbol vi ðxÞ represents the ground vibrational (ground electronic state), x0 is the vibrational frequency on the ground electronic state, eA is the energy difference between the excited state to which transition is allowed and ground electronic state while eF is the notation used for the forbidden electronic state. In a similar
Fig. 1. Schematic diagram showing the crossing of the diabatic potential energy curves illustrating the model.
Diwaker, A. Chakraborty / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 151 (2015) 510–514
513
profile significantly. However it is the Raman excitation profile that is more effected. 6. Conclusions In our earlier paper we have proposed an exactly analytically solvable model for the two state curve crossing problem discussing the case of crossing of two harmonic potentials. In this paper we have extended our approach to deal with the case of curve crossing induced dissociation by considering the crossing of a harmonic oscillator with a Morse potential with a possibility of wide applicability of our model. In the present manuscript we have analyzed the effect of curve crossing on electronic absorption spectrum and resonance Raman excitation profile and found that Raman excitation profile is affected much more by the crossing than the electronic absorption spectrum. Our method is very general and is applicable to arbitrary potentials as well as arbitrary couplings. Fig. 2. Calculated electronic absorption spectra showing the solid lines (with effect of coupling) and dashed lines (without effect of coupling).
Acknowledgments The author’s wants to thank IIT Mandi for Seed Grant as well as MHRD scholarships. Appendix A The general form of the diabatic potential matrix is given as
W DPM ¼
W 11 ðxÞ W 12 ðxÞ W 21 ðxÞ W 22 ðxÞ
;
ð27Þ
where 2
W 11 ¼ V 1 cos2 a þ V 2 sin a
ð28Þ
2
ð29Þ
W 22 ¼ V 1 sin a þ V 2 cos2 a and
W 12 ¼ W 21 ¼
Fig. 3. Calculated resonance Raman excitation profile showing the solid lines (with effect of coupling) and dashed lines (without effect of coupling).
ðV 1 V2Þ sin2a 2
ð30Þ
In the above expressions W 11 ; W 22 are the diabatic PEC while V 1 ; V 2 are the adiabatic PEC. W 12 ¼ W 21 is known as diabatic coupling and a is termed as mixing angle or the transformation angle. Further if W DPM represents the Diabatic potential matrix then it is related to the V APM (adiabatic potential matrix) by the following relation given as
V APM ¼ A ðaÞW DPM AðaÞ
ð31Þ
where
AðaÞ ¼
CosðaÞ SinðaÞ ; SinðaÞ CosðaÞ
ð32Þ
and
V ðAPMÞ ¼
Fig. 4. Variation of diabatic coupling with inter nuclear distance.
V1 0
0 ; V2
ð33Þ
we gave a generalized theoretical model which is applicable to any kind of molecular system. In any kind of realistic systems the diabatic coupling is not localized over R coordinate where R is the inter nuclear distance. To prove this we perform a test calculation on a certain diatomic molecule (LiCl) with the help of ab-initio studies (MOLPRO, a quantum chemistry package) and found that the diabatic coupling shows an approximate Gaussian type of nature with the inter nuclear distance as shown in figure below for the molecule under test calculation, hence it is not localized over nuclear coordinate. Our test calculation is not limited to only considered molecular system but it is applicable to any kind of molecular system. We
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Diwaker, A. Chakraborty / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 151 (2015) 510–514
include the following parameters to know about the nature of diabatic coupling for any kind of realistic system.
1 2 V 1 ¼ m x2 ðx d1 Þ 2
ð34Þ
with d1 ¼ 0 which represents the first harmonic oscillator near the minima of first PEC, and
V2 ¼
1 2 m x2 ðx d2 Þ 2
ð35Þ
with d2 ¼ 0:5 which represents the second harmonic oscillator near the minima of second PEC. For simplicity we took m = 1 and x = 1. The angle a which is known as mixing angle or transformation angle is obtained by performing an ab initio calculation for a test diatomic molecule over an internuclear distance ranging from (2 to 10) Å and thus diabatic coupling is obtained using Eq. (30). As per Refs. [34,35] in which it was mentioned that the diabatic coupling between the two diabatic curves is responsible for transitions between the two curves which will occur in a narrow range of nuclear coordinate given by
jW 11 ðxÞ W 22 ðxÞj ’ jW 12=21 ðxc Þj
ð36Þ
where xc is the crossing point. In reality the transitions between the two curves occur more effectively at the crossing point because the necessary energy transfer between the nuclear and electronic degrees of freedom is minimum at the crossing point, hence it is interesting to analyze a model where we have localized coupling instead of constant coupling or de localized coupling,hence we modeled this de localized coupling by Dirac Delta function and put our diabatic coupling as
W 12=21 ðxÞ ¼ K 0 dðx xc Þ
ð37Þ
and our diabatic potential energy matrix will look like as
W DPM ¼
W 11 ðxÞ W 12 ðxÞ W 21 ðxÞ W 22 ðxÞ
;
ð38Þ
where, W 11 = Harmonic Potential, W 12 = Morse Potential and diabatic coupling between the two is represented by
W 12 ðxÞ ¼ W 21 ðxÞ ¼ K 0 dðx xc Þ
ð39Þ
References [1] V.I. Osherov, Chem. Phys. Lett. 1 (1967) 539. [2] E.E. Nikitin, S.Ia Umanskii, in: M. Bayer, C.Y. Ng (Eds.), Theory of Slow Atomic Collisions, Springer, Berlin, 1984. [3] M.S. Child, Semiclassical Mechanics with Molecular Applications, Clarendon, Oxford, 1991. [4] H. Nakamura, Int. Rev. Phys. Chem. 10 (1991) 123. [5] H. Nakamura, in: R.E. Wyatt, J.Z.H. Zhang (Eds.), Dynamics of Molecules and Chemical Reactions, Marcel Dekker, New York, 1996. Chap. 12. [6] H. Nakamura, C. Zhu, Comments At. Mol. Phys. 32 (1996) 249. [7] S.S. Shaik, P.C. Hiberty, in: Z.B. Maksic (Ed.), Theoretical Models of Chemical Bonding, Part 4, vol. 82, Springer-Verlag, Berlin, 1991. [8] K. Nasu, Y. Kayanuma, Butsuri 35 (1980) 226 [In Japanese]. [9] F.O. Goodman, H. Nakamura, Prog. Surf. Sci. 50 (1995) 389. [10] Y. Gefen, E. Ben-Jacon, A.O. Caldeira, Phys. Rev. B 36 (1987) 2770. [11] E. Ben-Jacon, Y. Gefen, Phys. Lett. 108A (1985) 289. [12] A. Theil, J. Phys. G 16 (1990) 867. [13] N. Mataga, in: J.R. Bolton, N. Mataga, G. Mclendon (Eds.), Electron Transfer in Inorganic, Organic and Biological Systems, Advances in Chemistry, American Chemical Society, Washington DC, 1991. [14] B. Schwarzchild, Phys. Today (1987). [15] S.S. Shaik, J. Am. Chem. Soc. 103 (1981) 3692. [16] R. Kosloff, O. Citri, Faraday Discuss. 175 (1996) 96. [17] L.D. Landau, Phys. Zts. Sowjet 2 (1932) 46. [18] C. Zener, Proc. R. Soc. A 137 (1932) 696. [19] E.C.G. Stuckelberg, Helv. Phys. Acta 5 (1932) 369. [20] N. Rosen, C. Zener, Phys. Rev. 40 (1932) 502. [21] E.S. Medvedev, V.I. Osherov, Radiationless Transitions in Polyatomic Molecules, Springer, New York, 1994. [22] E.E. Nikitin, Annu. Rev. Phys. Chem. 50 (1999) 1. [23] H. Nakamura, Nonadiabatic Transition: Concepts, Basic Theories and Applications, World Scientific, Singapore, 2002. [24] H. Nakamura, in: M. Bayer, C.Y. Ng (Eds.), Theory, Advances in Chemical Physics, John Wiley and Sons, New York, 1992. [25] B. Imanishi, W. von Oertzen, Phys. Rep. 155 (1987) 29. [26] A. Yoshimori, M. Tsukada, in: A. Yoshimori, M. Tsukada (Eds.), Dynamic Processes and Ordering on Solid Surfaces, Springer-Verlag, Berlin, 1985. [27] R. Engleman, Non-Radiative Decay of Ions and Molecules in Solids, NorthHolland, Amsterdam, 1979. [28] D. Devault, Quantum Mechanical Tunneling in Biological Systems, Cambridge University Press, Cambridge, 1984. [29] S.H. Lin, Theor. Chim. Acta 1 (1967) 8. [30] M. Auznish, E.E. Nikitin, J. Troe, J. Chem. Phys. 103 (2008) 128. [31] N.L. Doltsinis, D. Marx, J. Theor. Comput. Chem. 1 (2002) 2. [32] M.D. Lecomte, B.L. Nihant, M.T. Praet, J.C. Lorquet, J. Chem. Phys. 7025 (1987) 86. [33] M.E. Alopyan, V.V. Baturo, J. Chem. Phys. 136 (2012) 234302. [34] A. Chakraborty, Mol. Phys. 107 (2009) 165. [35] A. Chakraborty, Mol. Phys. 107 (2009) 2459. [36] A. Chakraborty, Mol. Phys. 109 (2011) 429. [37] A. Chakraborty (Ph.D. Thesis), (Indian Institute of Science, India, 2004). [38] A. Chakraborty, Nano Devices, 2D Electron Solvation and Curve Crossing Problems: Theoretical Model Investigations, Lambert Academic Publishing, Germany, 2010. [39] Diwaker, A. Chakraborty, Mol. Phys. 110 (2012) 2257. [40] Diwaker, A. Chakraborty, Mol. Phys. 110 (2012) 2197. [41] C. Grosche, F. Steiner, Handbook of Feynman Path Integrals, Springer, 1998.
Contents lists available at ScienceDirect
Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy journal homepage: www.elsevier.com/locate/saa
Effect of curve crossing induced dissociation on absorption and resonance Raman spectra: An analytically solvable model Diwaker ⇑, Aniruddha Chakraborty Indian Institute of Technology Mandi, Mandi (H.P.) 175001, India
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
Model consisting of harmonic
oscillator and Morse potential is solved. Dirac Delta function and Greens function is used. Absorption spectra and resonance Raman excitation profile is studied.
a r t i c l e
i n f o
Article history: Received 17 April 2015 Received in revised form 27 June 2015 Accepted 1 July 2015 Available online 3 July 2015
a b s t r a c t An analytically solvable model for the crossing of a harmonic and a Morse potential coupled by Dirac Delta function has been proposed. Further we explore the electronic absorption spectra and resonance Raman excitation profile using this model and found that curve crossing had significant effect on the resonance Raman excitation profile. Ó 2015 Elsevier B.V. All rights reserved.
Keywords: Quantum mechanics Scattering Multi-channel Analytical model Greens function
1. Introduction In various atomic and molecular processes we have different type of transitions which occur between two atomic/molecular states and it is worthwhile to say that potential energy curve crossing plays a key role in such kind of transitions. Nonadiabatic transitions (transitions between two adiabatic states or diabatic states) generally occur between bound states [1]. Nonadiabatic transitions also play a very important role in change of state/phase [2–6] in different kind of dynamic processes occurring in the field of ⇑ Corresponding author. E-mail address: [email protected] ( Diwaker). http://dx.doi.org/10.1016/j.saa.2015.07.004 1386-1425/Ó 2015 Elsevier B.V. All rights reserved.
biology, chemistry and physics. Similarly, various spectroscopic, collisions processes and different kind of reactions are governed by nonadiabatic transitions at crossing or avoided crossing of the potential energy surfaces [7]. Radiationless transitions in condense matter physics, flouroscence quenching, self- trapping of excitons, laser assisted collisions reactions are some of the other examples [8,9] where nonadiabatic transitions due to curve crossing or avoided crossing play an important role. Other important fields of study related to nonadiabatic transitions include the Zener transitions in flux driven matellic rings [10], super conducting Josephson junctions [11], nuclear collisions and reactions in nuclear physics [12], electron proton transfer processes in biological molecules [13]. Such kind of transitions also play a very
511
Diwaker, A. Chakraborty / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 151 (2015) 510–514
important role in neutrino conversion in the sun and provide an evidence for its existence in the mutability of the universe [14]. It has been also commented by Shaik [15] that curve crossing/ avoided crossing could be interpreted as a general mechanism of rearrangement of electrons by molecules leading to the formation and breaking of bonds. Nonadiabatic transitions have been also studied as a mechanism of dissociation of molecules on metal surfaces as reported by Kosloff [16] in his studies. The first report on such kind of transitions came into existence into 1932, when the pioneering works were published by Landau [17], Zener [18] and Stueckelberg [19] and by Rosen and Zener [20]. Since1932, there are numerous research papers have been published in this area using analytical as well as computational approach [21–33]. In our earlier publications we have reported analytical solution in those cases where two or more arbitrary potentials are coupled by Dirac Delta interactions [34,37,35,38,36,39]. Recently we have extended our research to deal with the cases where two potentials are coupled by any arbitrary interaction [40]. One of the great advantage of Dirac Delta coupling model is that we can find exact analytical solution using this model.
H¼
H1 ðxÞ VðxÞ ; VðxÞ H2 ðxÞ
where Hi ðxÞ is
Hi ðxÞ ¼
1 @2 þ V i ðxÞ: 2m @x2
UðxÞ ¼
Z
1
ð1Þ
In the above equation x is the nuclear coordinate and xc is considered as crossing point. V 1 ðxÞ and V 2 ðxÞ are the diabatic potentials (and Vðxc ) represent the coupling between them. In real systems the transitions between the two diabatic potentials occurs at the crossing point. This can be attributed to the fact that necessary energy transfer between the electronic and nuclear degrees of freedom is minimum at this point, hence it is worthful to analyze a coupling which is localized near the crossing point rather than using a model where coupling is constant or same elsewhere. Thus we put (see Appendix A for more details on diabatic coupling, where we do a calculation on real system like Lithium chloride i.e. (Eq. (2), Fig. 4))
VðxÞ ¼ k0 dðx xc Þ;
3. Exact analytical approach In this section we derive exact analytical expressions for wave function and Green’s function which are needed to study the effect of curve crossing on electronic absorption and resonance Raman excitation profile. We write the probability amplitude for the two included states as
/1 ðx; tÞ /2 ðx; tÞ
;
ð3Þ
where /1 ðx; tÞ and /2 ðx; tÞ are the probability amplitude for the two states. Uðx; tÞ obey the time dependent Schrodinger equation. In the h ¼ 1) subsequent calculations as well as the present one (we take
i
@ Uðx; tÞ ¼ HUðx; tÞ: @t H is given by
ð4Þ
ð7Þ
writing the half Fourier transform of Eq. (4) gave us
UðxÞ ¼ iGðxÞUð0Þ;
ð8Þ
where GðxÞ is defined by
ðx HÞGðxÞ ¼ I:
ð9Þ
In the position representation, the above equation may be written as
Z
1
Gðx; x0 ; xÞUðx0 ; xÞdx0 ;
ð10Þ
1
where Gðx; x0 ; xÞ is
Gðx; x0 ; xÞ ¼ hxjðx HÞ1 jx0 i:
ð11Þ
Writing
Gðx; x0 ; xÞ ¼
G11 ðx; x0 ; xÞ G12 ðx; x0 ; xÞ
ð12Þ
G21 ðx; x0 ; xÞ G22 ðx; x0 ; xÞ
and applying the partitioning technique we can write 1
G11 ðx; x0 ; xÞ ¼ hxj½x H1 Vðx H2 Þ1 V jx0 i:
ð13Þ
The above equation is true for any general potential VðxÞ. Our equation is further simplified if V is a delta function located at the crossing point
G11 ðx; x0 ; xÞ ¼ G01 ðx; x0 ; xÞ þ
ð2Þ
here k0 is a constant. This model has the advantage that it can be analytically solved.
Uðx; tÞ ¼
UðtÞeixt dt:
0
Uðx; xÞ ¼ i
jV 1 ðxÞ V 2 ðxÞj ’ jVðxc Þj:
ð6Þ
the half Fourier transform UðxÞ of UðtÞ is defined by
2. Mathematical model We consider the crossing of two diabatic curves (a harmonic potential and a Morse potential in the present case) and the coupling between them is assumed to be a Dirac Delta function which is responsible for transitions between these two considered diabatic curves. Nonadiabatic transition due to curve crossing would occurs in the neighborhood of the crossing point or in other words is given by
ð5Þ
K 20 G01 ðx; xc ; xÞG02 ðxc ; xc ; xÞG01 ðxc ; x0 ; xÞ 1 K 20 G01 ðxc ; xc ; xÞG02 ðxc ; xc ; xÞ
;
ð14Þ
where
G0i ðx; x0 ; xÞ ¼ hxjðx Hi Þ1 jx0 i;
ð15Þ
The above value of Green’s function corresponds to propagation of the particle starting at x0 on the second diabatic curve, in the absence of coupling to the first diabatic curve. In a similar fashion one can get
G12 ðx; x0 ; xÞ ¼
K 0 G01 ðx; xc ; xÞG02 ðxc ; x0 ; xÞ 1 K 20 G01 ðxc ; xc ; xÞG02 ðxc ; xc ; xÞ
:
ð16Þ
The expressions for G22 ðx; x0 ; xÞ and G21 ðx; x0 ; xÞ can be further derived by using a similar approach as shown in the last section. We can calculate UðxÞ explicitly by using these expressions for the Green’s function in Eq. (8). The expressions that we have obtained for UðxÞ are quite general and are valid for any V 1 ðxÞ and V 2 ðxÞ. However, their utility is limited by the fact that one must know G01 ðx; x0 ; xÞ and G02 ðx; x0 ; xÞ. It is possible to find G0i ðx; x0 ; xÞ only in a few limited cases and the Morse oscillator is one of them [41].
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4. Curve crossing induced dissociation: electronic absorption and resonance Raman excitation profile In the present section the mathematical model along with necessary derivations derived above have been used to solve the problem involving a harmonic oscillator and a Morse oscillator, coupled by Dirac Delta function. A system involving three Potential energy curves has been considered out of which we have a ground electronic state and two ‘crossing’ excited electronic states. Electronic transition to one of them is assumed to be dipole forbidden and while it is allowed to the other. We calculate the effect of ‘crossing’ induced dissociation on electronic absorption spectra and on resonance Raman excitation profile. The propagating wave functions on the excited state potential energy curves are given by solution of the time dependent Schrödinger equation
@ i @t
U1v ib ðx; tÞ
!
v ib
U2 ðx; tÞ
¼
Hv ib;e1 ðxÞ
V 12 ðxÞ
V 21 ðxÞ
Hv ib;e2 ðxÞ
U1v ib ðx; tÞ v ib
U2 ðx; tÞ
! :
ð17Þ
In the above equation Hv ib;e1 ðxÞ and Hv ib;e2 ðxÞ describes the vibrational motion of the system in the first electronic excited state which is allowed and second electronic excited state which is forbidden
Hv ib;e1 ðxÞ ¼
1 @2 1 þ mx2A ðx aÞ2 2m @x2 2
ð18Þ
1 @2 2 þ DF ½1 eðxbÞ : 2m @x2
ð19Þ
and
Hv ib;e2 ðxÞ ¼
In the above m is the oscillator’s mass, xA is the vibrational frequency of the first electronic excited state and DF is the dissociation energies of the forbidden states and x is the vibrational coordinate. Shifts of the vibrational coordinate minimum upon excitation are given by a and b, and V 12 ðxÞ (V 21 ðxÞ) represent coupling between the two diabatic potentials which is taken to be
V 21 ðxÞ ¼ V 12 ðxÞ ¼ K 0 dðx xc Þ;
ð20Þ
where K 0 represent the strength of the coupling. The equation representing the intensity of electronic absorption spectra can be written as
Z IA ðxÞ / Re
dx0 Uvi ib ðxÞiGðx; x0 ; x þ iCÞUiv ib ðx0 Þ ;
ð21Þ
E D 1 Gðx; x0 ; x þ iCÞ ¼ x½ðx0 =2 þ x xeg Þ þ iC Hv ib;e x0 :
ð22Þ
1
dx
Z
1
1
1
way we can write the equation representing resonance Raman scattering intensity in terms of Green’s function and is given by.
Z IR ðxÞ /
1
dx
1
Z
1 1
2 dx0 Ufv ib ðx; 0ÞiGðx; x0 ; x þ iCÞUiv ib ðx0 ; 0Þ : ð25Þ
In the above Ufv ib ðx; 0Þ is given by
Ufv ib ðx; 0Þ ¼
vf ðxÞ 0
;
ð26Þ
vf ðxÞ is the final vibrational state of the ground electronic state. As G0i ðx; x0 ; xÞ for the harmonic potential is known, we can calculate Gðx; x0 ; xÞ. We use Eq. (25) to calculate the effect of curve where
crossing induced dissociation on resonance Raman excitation profile.
5. Results In the following section we discussed the results of curve crossing on both the electronic absorption spectra and resonance Raman excitation profile as shown in Fig. 1. The ground state curve is considered to be a harmonic curve with its minimum at zero. This curve can be exactly mapped to the case of potential energy curve produced in stretching of a metal ligand bond. We took mass as 35 a.m.u. in our calculations and 400 cm1 as the vibrational wavenumber for the ground state. The first diabatic excited state to which transitions are allowed is displaced by 0.1 Å. Its minimum is taken to the above 10,700 cm1 in comparison to ground state curve. The same parameters for another diabatic curve to which transitions are forbidden are taken as 400 cm1 and 10,800 cm1 respectively. These two excited diabatic curves crosses at an energy of 10; 804:1 cm1 . Value of xc and k0 in our calculations are 0:02477 Å and 5:54277 1015 erg Å respectively. 450 cm1 is taken to be as life time of both the excited states. The result of our calculations are shown in Figs. 2 and 3 respectively. The profile shown by the dashed line is in the absence of any coupling to the second potential energy curve while the profile shown in solid line has the effect of coupling in it. It is seen that curve crossing effect can alter the absorption and Raman excitation
where
and
Hv ib;e ¼
Hv ib;e1 ðxÞ
K 0 jxc ihxc j
K 0 jxc ihxc j
Hv ib;e2 ðxÞ
ð23Þ
Here, C is a phenomenological damping constant which account for the life time effects. Uiv ib ðx; 0Þ is given by
Uvi ib ðx; 0Þ ¼
vi ðxÞ 0
;
ð24Þ
where the symbol vi ðxÞ represents the ground vibrational (ground electronic state), x0 is the vibrational frequency on the ground electronic state, eA is the energy difference between the excited state to which transition is allowed and ground electronic state while eF is the notation used for the forbidden electronic state. In a similar
Fig. 1. Schematic diagram showing the crossing of the diabatic potential energy curves illustrating the model.
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513
profile significantly. However it is the Raman excitation profile that is more effected. 6. Conclusions In our earlier paper we have proposed an exactly analytically solvable model for the two state curve crossing problem discussing the case of crossing of two harmonic potentials. In this paper we have extended our approach to deal with the case of curve crossing induced dissociation by considering the crossing of a harmonic oscillator with a Morse potential with a possibility of wide applicability of our model. In the present manuscript we have analyzed the effect of curve crossing on electronic absorption spectrum and resonance Raman excitation profile and found that Raman excitation profile is affected much more by the crossing than the electronic absorption spectrum. Our method is very general and is applicable to arbitrary potentials as well as arbitrary couplings. Fig. 2. Calculated electronic absorption spectra showing the solid lines (with effect of coupling) and dashed lines (without effect of coupling).
Acknowledgments The author’s wants to thank IIT Mandi for Seed Grant as well as MHRD scholarships. Appendix A The general form of the diabatic potential matrix is given as
W DPM ¼
W 11 ðxÞ W 12 ðxÞ W 21 ðxÞ W 22 ðxÞ
;
ð27Þ
where 2
W 11 ¼ V 1 cos2 a þ V 2 sin a
ð28Þ
2
ð29Þ
W 22 ¼ V 1 sin a þ V 2 cos2 a and
W 12 ¼ W 21 ¼
Fig. 3. Calculated resonance Raman excitation profile showing the solid lines (with effect of coupling) and dashed lines (without effect of coupling).
ðV 1 V2Þ sin2a 2
ð30Þ
In the above expressions W 11 ; W 22 are the diabatic PEC while V 1 ; V 2 are the adiabatic PEC. W 12 ¼ W 21 is known as diabatic coupling and a is termed as mixing angle or the transformation angle. Further if W DPM represents the Diabatic potential matrix then it is related to the V APM (adiabatic potential matrix) by the following relation given as
V APM ¼ A ðaÞW DPM AðaÞ
ð31Þ
where
AðaÞ ¼
CosðaÞ SinðaÞ ; SinðaÞ CosðaÞ
ð32Þ
and
V ðAPMÞ ¼
Fig. 4. Variation of diabatic coupling with inter nuclear distance.
V1 0
0 ; V2
ð33Þ
we gave a generalized theoretical model which is applicable to any kind of molecular system. In any kind of realistic systems the diabatic coupling is not localized over R coordinate where R is the inter nuclear distance. To prove this we perform a test calculation on a certain diatomic molecule (LiCl) with the help of ab-initio studies (MOLPRO, a quantum chemistry package) and found that the diabatic coupling shows an approximate Gaussian type of nature with the inter nuclear distance as shown in figure below for the molecule under test calculation, hence it is not localized over nuclear coordinate. Our test calculation is not limited to only considered molecular system but it is applicable to any kind of molecular system. We
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include the following parameters to know about the nature of diabatic coupling for any kind of realistic system.
1 2 V 1 ¼ m x2 ðx d1 Þ 2
ð34Þ
with d1 ¼ 0 which represents the first harmonic oscillator near the minima of first PEC, and
V2 ¼
1 2 m x2 ðx d2 Þ 2
ð35Þ
with d2 ¼ 0:5 which represents the second harmonic oscillator near the minima of second PEC. For simplicity we took m = 1 and x = 1. The angle a which is known as mixing angle or transformation angle is obtained by performing an ab initio calculation for a test diatomic molecule over an internuclear distance ranging from (2 to 10) Å and thus diabatic coupling is obtained using Eq. (30). As per Refs. [34,35] in which it was mentioned that the diabatic coupling between the two diabatic curves is responsible for transitions between the two curves which will occur in a narrow range of nuclear coordinate given by
jW 11 ðxÞ W 22 ðxÞj ’ jW 12=21 ðxc Þj
ð36Þ
where xc is the crossing point. In reality the transitions between the two curves occur more effectively at the crossing point because the necessary energy transfer between the nuclear and electronic degrees of freedom is minimum at the crossing point, hence it is interesting to analyze a model where we have localized coupling instead of constant coupling or de localized coupling,hence we modeled this de localized coupling by Dirac Delta function and put our diabatic coupling as
W 12=21 ðxÞ ¼ K 0 dðx xc Þ
ð37Þ
and our diabatic potential energy matrix will look like as
W DPM ¼
W 11 ðxÞ W 12 ðxÞ W 21 ðxÞ W 22 ðxÞ
;
ð38Þ
where, W 11 = Harmonic Potential, W 12 = Morse Potential and diabatic coupling between the two is represented by
W 12 ðxÞ ¼ W 21 ðxÞ ¼ K 0 dðx xc Þ
ð39Þ
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