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On the theory and MD-simulation of one-photon electronic excitation of multy-atomic molecules
B. A. Grishanin*), V. D. Vachev, V. N. Zadkov

International Laser Center, Moscow State University Lenin's Hills, 119899 Moscow, USSR *)physics Department, Moscow State University Lenin's Hills, 119899 Moscow, USSR

ABSTRACT
Quantum description of the molecular dynamics under laser excitation of electronic transitions is reduced to an equivalent classical form. This classical treatment is applied to the molecular dynamics problem and some specific results for one-dimensional case are demonstrated. Absorption probability and laser induced quantum coordinate displacement spectra are calculated.

1. INTRODUCTION
2 are in the evident contradiction with the quantum peculiarity of laser molecule exciClassical MD tation. Quantum methods are necessary for description of the initial stage of excitation. If the first stage of the
excitation and further evolution can be described separately, classical MD-analyses of further evolution can be employed (Fig. 1). It has been also demonstrated that laser pulse in a two-level electronic molecular system excites essentially delocalized rovibronic states only if pulse duration is not extremely short.3 A good number of calculations of excited states have been produced in a fully quantum manner by computation of the wave functions. This approach however is good only for small molecules and is not effective for multi-atomic molecules which are also of great interest now for example, for biomacromolecules such as rhodopsin, bacteriorhodopsin,4 stilbene.5

In this work we present a partially analytical way of calculating of excited states characteristics of harmonic systems. The Gaussian operator form technique6 is used for the representation of quantum dynamics as oscillations of classical coordinate-momentum pair X = (x, p). Thus we have classical concept that is completely adequate to the quantum description and it can put in accordance the quantum and classical concepts.

2. BASIC FORMULAE
Excited state can be described by the formula of the first order perturbation theory respective to the laser field for the quantum density matrix of the upper electronic state 2):
P22
where

di2

[f

dr1 dr2EL(T,)E(r2)e_2(t s)jjO(r)e2(t s)

(1)

r = T2 -- Ti, S = (r2 + r,)/2, EL(Tk) is a complex wave function of the laser field with amplitude EL, d,2 is an electronic transition dipole moment, h is the Plank constant;
= e_22n/'2 e1T/2 ,f, e1T/2 e'52'hi2
(2)

is generalized density matrix of the vibronic state that excited at a medium moment s starting from the ground electronic state 1) and then being formed during time interval r by combining dynamics of the both terms 1), 2); ,5f, is initial ground equilibrium state with a temperature T;

Ж,,2= [l3Tm_ll3/2+U,2(1)I/1

(3)

44 / SPIE Vol. 1402 USSR-CSFR Joint Seminar on Nonlinear Optics in Control, Diagnostics, and Modeling of Biophysical

Processes (1990)


classical MD--simulation

'

>
w
LiJ

initial excited state distribution

I

12)

quantum harmonic theory classical
MD--simulation
i-...

quantum or phenomenological theory

Ii>

initial ground state distribution

x
Fig. 1. Stages of MD calculations of molecule dynamics under electronic excitation by laser pulse (L'L is a laser frequency).

SP/E Vol. 1402 USSR-CSFR Joint Seminar on Nonlinear Optics in Control,

Diagnostics, and Modeling ofBiophysica/ Processes (1990) / 45


are molecular terms energy operators in frequency units. Due to the harmonicity of both terms expression (2) determines the generalized Gaussian density matrix (see below) and so the correspondent state can be classicaly interpreted as a result of oscillations of the classical coordinates x and momenta p, initially uncertained because of vacuum and temperature fluctuations given by the Plank formula. Any characteristics of this state may be
represented analytically by proper matrix expressions using the frequency and mass matrices for the both molecular terms. In this article we present only calculations of average values of density matrix (2) itself and coordinatemomentum pair z , p.

i -- S.

In expression (1) the transformation e2'2(t _ s) determines the free evolution of the upper state by the time This evolution is nothing but classical harmonic oscillations of the upper term only. It may be omitted if

the final result depends on the evolution at time intervals, much greater than a laser pulse duration r,. Sometimes it is possible for a certain class of problems studied by MD such as isomerization. For the purpose of absorption spectrum calculations it is always possible because of the conservation of the excited state probability at short times
i f.- Tp.

In these cases the subject of interest is the effective density matrix
P22

IdiI2
JJdnldT2E1(nl)EL(T2)P22(T).

It

is initial density matrix for MD problems connected with the excited state. It is the integral of Gaussian generalized density matrix j32(T) if the ground state is Gaussian. It differs mainly from the ground state by average values of X1 due to the photon energy excess hL -- (U2 -- U1), where Ui,2 is calculated for i = xi (the coordinates Xq potential energy minimum point of the ground state). So, the full set of the j32-matrix characteristic parameters
are: P(WL) -- quantum transition probability to the excited state 2); &(r) -- instantaneous mean displacement of X;

Xq (xq, pq) -- average value ofX over pulse duration; -- Zt)I) -- instantaneous correlation matrix corresponding to 32(r); K(r) =
K--
K= x: -- full correlation matrix including fluctuations of displacements,K +
average

correlation matrix, produced as a result of averaging K(r);

(zx(r)zx(r)).

Using these data one can assosiate exact MD analyses of molecules excited by a laser pulse. It is useful to note is not the eigen state for Hamiltonian 71/62 even for that in contrast to real density matrix j522, the effective one the stationary conditions at the exact resonance (as in Ref. 3). Somehow it not worth while marking specially the stationary states of the excited level in case of short picosecond pulses.

3. RESULTING FORMULAE
Calculating the value Trf°2 for I >> r, we obtain the following expression for transition probability
P(WL) =

(l/4)
62

J drf(r) exp {i[wL - (2

- )]r}

x(r),

(5)

where lL = EL d12 is the Rabi frequency; f(r) = dsu(s -- r/2) u(s + r/2) is the autocorrelation function of
complex normalized amplitude u; c5i,
-- x(r) --

are calculated in the point x1.
det

f

[sinh () (shi sh2Y1]
(cthi/wi + cth2/w2)' mh/2 (2 --

det"2(cthi/wi + cth2/w2) det112(cth1w1 + cth2w2)
(x2 -- xi)Tmh/2

x exp

x1)}

(6)

46

/ SPIE Vol. 1402 USSR-CSFR Joint Seminar on Nonlinear Optics in Control, Diagnostics, and Modeling of Biophysical Processes (1990)


is a generalized function of transition probability for time t while photon excitation of upper term 2) is being energy minimum point of the term 2), m is 3N x 3N-mass-matrix if the considered system is Nformed, x2
atomic molecule;'
sh, = sinh [w,(ti/kT -- ir)/2], sh2 = sinh (iw2r/2), ch2 = cosh (iw2r/2), cth, = coth [w,(fl/kT -- ir)/2], cth2 = coth (iw2r/2)
are complex or pure imaginary expressions with w ,

2) 2 5 imaginary) and k is the Boltzman constant.
(LXvac

2

-- frequency matrices for terms 1) 12) (for dissociative term

This generalized probability is not actually positive despite the classical character of initial vibrational state O.1A) because of actually quantum interaction with the electron states, but it is definitely positive as an operator kernel and its Fourier transformation is always positive. These resulting formulae give the exact onephoton expression and include vacuum shifts due to vacuum term energies 1k4,/2, 11w2/2 in contrast to a semiclassical statistical approach including only the exponential factor of Eq. (6).

For average x-coordinate we have:
Xq X2

+JP(T)LX(r)dT/P(WL),

(7)

where p(r) is time density of P(c4L) (see Eq. (5));

zx(r) = --m"2w'sh'(cth,/w, + cth2/w2)'m'/2(x2 --x,)
is the displacement at the moment r respectively to x2. For average momenta we have Pq
correlation matrix K(r) is given by the following expression:

(8)

0. Instantaneous

K"r -- (K(r) ' J--k\ 0
where

Kp(r))'

0\

K(r) = m_h/2(cth,/w,w2sh2 + ch2)'(sh2/w2 + cth,/w, ch2)rn'/2,
K(r) = m1/2(cth,w,/w2sh2 + ch2)'(sh2w2 + cth,w,ch2) rn"2.

(9) (10)

Eqs. (7)--(10) describe the superposition of the quantum initial fluctuations and the free classical vibrations in the ground state Ii) with the following classical vibrations in upper state 2). Using these formulae we can calculate all characteristics of generalized density matrix (2) as for Gaussian case they all can be expressed by resulting iXx(r)

and K(r).

In one-dimensional case the mean values Xq, K for distribution (4) may be easily expressed by the combinations of values P(WL nw1 1w2) due to the periodicity of the corresponding time dependencies. For the case T = 0 and = w2 = w after integration over time we have
Xq --

=

(x2 --

x,)

P(WL) -- P(wL

--

w/2)

(11)

Zero subspaces due to the conservation of transitional and rotational integrals must be excluded but corresponding spectrum effects can also be taken into account after adding proper terms.

SPIE Vol. 1402 USSR-CSFR Joint Seminar on Nonlinear Optics in Control, Diagnostics, and Modeling of Biophysical Processes (1990) / 47


This expression gives zero displacement for frequencies WL Wm + w/4 and infinite one for zero absorption. For the correlation matrix at T = 0 we have the following expression:
Ii = m_ _
kfl(w2/1)

P(wL)--P(wL--n2)
,

(12)
(13)

=
and k --

71

h

mw1

k(w2/w1)

,

P(WL)

--

P(WL

_

nw2)

,

where k is the Fourier-coefficients for periodic function 1(r)

for function 1/1(r). The corresponding instantaneous fluctuations K(r), K(r) are vacuum ones

= (1+wi/w2th2) (1+w2/with2)', th2 = th (i2 r/2),

but they do not correspond to the vacuum state of term 2). They are oscillating in a complex region with the values h/(2rnwi), Ilmwi/2 at r = 0 and liwi/(2rm4), 11mw/(2mwi) at r = ir/w2; minimum of the Heisenberg uncertainty conserved but the proper states in a general case are squeezed. These latter equations are valid only

for zero temperature, for which the initial states before excitation are illustrated by Fig. 2a. For T 0 we
can simplify the interpretation of the phenomenon using the representation of T-fluctuations as a result of random displacements of vacuum state (see Fig. 2b) . So, in the following operator equations we must use representation j3f =

fpr(X)Uxj31Uj' dX with the Gaussian distribution of temperature fluctuation p'r(X) and shift transformation

U.

The described formulae are the corollary of the the well-known composition rule for operators X and Gaussian exponents:

Xe =
C=

where k is a vector set of operators X with scalar commutator

(JJ -

For N-atomic molecules X is a 6N-vector and Q, C are 6N x 6N-matrices in a vector space of phase coordinates X. For zX(r) = TrXp2(r)/(r) -- X1. Let us represent follows:

ji(r) =
where Ui , U2, ,T1 are

U2(r)Ui(r)i3'1Ui(r)U2(r),

the Gaussian operator forms. Using the above commutation rule we deduce the middle-stage equation with appropriate quadratic form matrices Qi , Q2 corresponding to electronic terms 1) 12): zX(r) = (e2C22 e2CQ1

e2CQ2 i)' (e2C2

e2C1 + 1) (e2CQ2

1) (X2 --

X1).

After some proper transformations we obtain Eqs. (6), (7) and get the expression

K(r) = (e' e2C2 e_' e_22) (e' e22 + e'?1 e_22)
To calculate the absorption probability we can use the following expression:

C.

x(r) =
with corresponding a, Qi, Q2. Using also characteristic function technique6 we obtain Eq. (5) if in addition to the above we take into account the impossibility to represent the normalization constants as a modula. Using also the technique of Wigner representation we can also get the analytical or computer usable representation of density matrix j5 itself. However, it needs using a good number of rather complicated but routine-like transformations of quadratic forms. At the same time many real calculations may be performed without using this representation.

48

/ SPIE Vol. 1402 USSR-CSFR Joint Seminar on Nonlinear Optics in Control, Diagnostics, and Modeling of Biophysical Processes (1990)


potential surface
vacuum coordinate distribution

potential surface with kinetic energy

>
U)

/

/

I

w
..... : : : :·..·.

(a)

xi

x
potential surface

potential surface with kinetic energy

>
w
LiJ

txO
x=O

(b)

Fig. 2. The structure of the vibronic distribution of ground state: (a) -- vacuum state (T = 0), (b) -- temperature equilibrium state (T 0).

x

SPIE Vol. 1402 USSR-CSFRJoint Seminar on Nonlinear Optics in Control, Diagnostics, andModeling ofBiophysicalProcesses (1990) / 49


4. COMPUTER SIMULATION AND DISCUSSION
In this work we have analyzed the excitation of a one-dimensional harmonic system with two different harmonic terms as a simple model of a real molecule. Calculations for real multi-atomic molecules do not require any special methods and are in progress now. Figures 3--6 show typical spectra of absorption probability in comparison with

the same ones for the electronically equal two-level atom and the spectra of x-coordinate displacement for the Gaussian pulse with r = 0.5 p5; electronic transition frequency w12 = 1015 s1; mass parameter m = 1.4 x 1O_23 g; vibronic frequency for the ground state w1 = 1013 1 and for the excited state w2 1013, 3 x 1013, 5 x 1013 5_i; temperatures T = 0, 300 K; terms displacements x2 -- xi = 0, O.3A.
We can see the following properties of spectra on Figs. 3--6: -- the partially resolved satellite structure of spectra; quantum vacuum shift (w2 -- wi)/2 (w2 = 5 x 1O'3s') and vibronic resonances on additional frequencies for X2 -- X1 0 at T 0 K (Fig. 3); -- the Condon frequency shift for x2 -- x1 = 0.3 A (Figs. 4, 5, 6); -- the vanishing of resonances at the frequencies below the electronic resonance for the value of nwi at wi =

andT=O(Fig. 4);

-- simmetrization of spectral lines at the above frequencies at temperature T = 300 K (Fig. 6).

The formation of the probability of the electronic transition is reversible because of the absence of statistical averaging (the entropy of vacuum state is equal to zero). So, we can see the vibronic resonances not only above the electronic one. This is connected with the finite duration of laser pulse. The coordinate displacement Xqwhich is proportional to x2 -- x1 shows the large maxima at the points of minimum absorption.
The above mentioned rule of zero displacement (see Eq. (11) and the text below) is confirmed by Figs. 4--6. For the laser frequencies below the resonance relation Xq > 0 is more typical and at the laser frequencies above the

resonance we have Xq < 0 (Fig. 4). It is correlated with the main direction of coordinate shift on the potential surface due to the excess of the photon energy (see Fig. 1). In the region of meaningful absorption (Figs. 4, 6). the typical value of the displacement zx(r) is about 1x2 -- x1 But it can be infinitely large in the regions of weak absorption (Fig. 5).

.

For one-dimensional system the computation of one spectral curve on the personal computer IBM P5/2-50 takes about 1 minute. In many-dimensional systems there are two extremely different cases: commuting (Wi W2 -- W2W1 = 0) and non-commuting frequency matrices. For the first one, the calculations based on Eq. (5) do not require independent spectral analyses for different r because of simultaneous diagonalization of the both matrices wi , W2. So, the number of operations increases in 3N times for the first case and in 3N x 3N times for the second one.
Therefore, for multi-atomic molecules with terms differing only in displacement x2 -- , the calculations described above are really realized by modern computers. There exist a general class of molecules with the commuting frequency

x

matrices of electronic terms. It is represented by molecules with terms differing not only in displacement but in additional scaling transformation x' -- x2 = a(x -- x1) as well. It means that the values of force constants for the both terms are proportional. Sometimes it can be accepted as a good approximation.

5. CONCLUSION
The methods presented in this article based on analytical and classical approach and further computer simulation are applicable for MD analyses of laser excited multi-atomic molecules.

6. ACKNOWLEDGEMENT
This work was partially supported by the Program "Engineering Ensimology" Grant #1-250.

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/ SPIE Vol. 1402 USSR-CSFR Joint Seminar on Nonlinear Optics in Control, Diagnostics, and Modeling of Biophysical Processes (1990)


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7. REFERENCES
1. J. R. Beeler (Jr), Physics of many particle systems, Ed. C. Meeron, N.Y., Gordon & Breach, 1964. 2. D. W. Heerman, Computer Simulaiion methods in theoretical physics, Springer, Berlin, 1990. 3. M. V. Rama Krishna and R. D. Coalson, Chem. Phys., 120, p. 327, 1988. 4. R. R. Birge and L. M. Hubbard, Biophys. J., 34, p. 517, 1981; R. R. Birge, L. A. Findsen and B. M. Pierce, J. Am. Chem. Soc., 109, p. 5041, 1987. 5. V. D. Vachev, V. N. Zadkov, SPIE Proc., 1402, 1991. 6. B. A. Grishanin, Quanium elecirodynamics for radiophysisiss, MSU, Moscow, 1983 (in Russian). 7. E. J. Heller, Accounts Chem. Res., 14, p. 386, 1981. 8. B. A. Grishanin, V. M. Petnikova and V. V. Schuvaloff, Soviet J. Appl. Spectr., 47, p. 386, 1987 (in Russian).

52 / SPIE Vol. 1402 USSR-CSFR Joint Seminar on Nonlinear Optics in Control, Diagnostics, and Modeling of Biophysical Processes (1990)