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Hidden crossings and non-adiabatic transitions between rotational states of H2O in a slow varying external field
23 November
1998
PHYSICS
ELSEVIER
Physics Letters A 249 (1998)
LETTERS
A
74-76
Hidden crossings and non-adiabatic transitions between rotational states of Hz0 in a slow varying external field E.A. Solov’ev Macedonian
Academy of Sriences and Arts. PO. Box 428, Skopje, Macedonia
Received 20 July 1998; accepted Communicated
for publication by B. Fricke
17 August
1998
Abstract The hidden crossings of rotational energy levels of a Hz0 molecule embedded in an electric field are studied. It is found that non-adiabatic transitions associated with the hidden crossings become significant if the product FOO 2 lo-*, where FO is the amplitude of the periodic field and w is its frequency (both in atomic units). @ 1998 Elsevier Science B.V.
In a simple approximation the transitions between rotational states induced by the interaction of an electric dipole of a molecule H20 with a time-dependent external field are described by the Schrodinger equation t (see, e.g., Ref. [ 1I )
(
-&L’-F(ot).d @(ii,t)
=i
a$$,
at
r>
$<;i,t)
=
cp,<&F(d)) t
x exp
E,( F( ot’) dt’
-i (
s
>
,
(2)
where (~,(a, F) and E,(F) are the eigenfunctions and eigenvalues of the instantaneous Hamiltonian, 9 (1)
>
where F is an external electric field, o is the rate of changing of the electric field, which further is assumed to be a small parameter in the adiabatic approximation, L, I and d are the angular momentum, momentum of inertia and electric dipole vector of the molecule (in the case of HZO: I = 5692 a.u., IdI = 0.756 a.u. [l]), 3 = d/d determines the direction of the dipole vector. In the leading order of the adiabatic approximation the solution of Eq. ( 1) reads
] Atomic units are used throughout
this work.
- $Lz
- Fad
>
(o&F)
=E,(F)po,(&F), (3)
cy is the set of quantum numbers of eigenstates. Since in the limit F -+ 0 the eigenfunctions are transformed into spherical functions, we will specify the adiabatic states by their spherical quantum numbers at F = 0, i.e. ff = llm] . In the frame of reference with the z-axis directed along the external field F, Eq. (3) admits separation of the variables in spherical coordinates 6, 4. Substitution of the wavefunction in the form cpa(d, F) ) = problem V(8) e impleads to the one-dimensional
03759601/98/$ - see. front matter @ 1998 Elsevier Science B.V. All rights reserved. PIISO375-9601(98)00649-S
E.A. Solov’evlPhysics
Letters A 249 (1998)
74-76
15
Re(F)xlO’ (a.u.) Fig. 2.
where 0.5
1.5
1.0
Fx103
2.0
(au)
Fig. 1.
1 d . dV(@ -sm(8)dB sin(O) d6’ +
(
2ZE,(F)
=o.
- 21Fdcos(8)
m2 - ~ V(O) sin’( 0) > (4)
Fig. 1 shows adiabatic energy curves with 1 < 4 and m 6 2. The calculation has been done using a similar code as previously developed for the two-Coulomb center problem [ 2,3]. The adiabatic approximation is based on the fact that the set of energy curves E,(F) of the same symmetry (in our case, having the same value m) are different sheets of a single analytical function E(F); they are connected pairwise (Ia) - I/3)) at certain branch points Fap in the complex F-plane. In this approach 2 , the transition between two adiabatic states ]LY)and I/?) can be obtained as a result of analytical continuation of the wavefunction (2) along the contour Cap (in the complex r or F planes, which are connected analytically) enclosing the related branch point Fmp (Zwaan’s method [ 51) . This procedure gives the following expression for the amplitude of transition, Aap = e - d”p
(5)
‘The adiabatic approach is described in detail, for instance, in review 14)).
The exact crossing of energy surfaces E,(F) and Z+(F) at the branch point F+ reflects the strong coupling of the adiabatic states at the real axis F caused by two different mechanisms: Landau-Zener avoided crossings and hidden crossings. Initially hidden crossings between adiabatic states have been revealed in slow atomic collisions [ 3] . The hidden crossings appear whenever the classical trajectory related to the adiabatic state collapses into an unstable periodic orbit. Then the intensity of the transition is determined by the Lyapunov exponent of this orbit [ 61. The hidden crossings are alternatives to the well-known Landau-Zener avoided crossings, which obey the resonant underbarrier interaction of adiabatic states located in different potential wells. In contrast to the Landau-Zener crossings, the hidden crossings always exist and provide a full description of the non-adiabatic transitions. The characteristic of hidden crossings is that they are not distinguished in the conventional pattern of adiabatic energy curves at a real value of a slow varying parameter and their description requires direct computation in the complex plane of a parameter. Fig. 2 illustrates the distribution of the branch points F,p in the complex F-plane for the set of energy curves in Fig. 1. The complex quantity Fop is of major practical importance. Its real part (Re F,p) corresponds to the strength of the electric field at
76
E.A. Solov’ev/Physics
Letters A 249 (1998) 74-76
Table 1 Branch points F,+q (multiplied by 103) and parameters &in (multiplied by 106)
1:
I4
IP)
Re F,p
Im Fma
&in
[Lull 10.01
[I+ I,tn] Il.01
x10” (a.u.) 0
x IO3 (au) 0.2181081
x IO6 0.0030240
bF
I-
Il.01
I2,Ol
0.3824243
0.5483754
0.1444546
2
[3*01 14.01 12.11
1.0638385
0.9025318
*Qg
2.0355 I97 0
0.6216384
0.3379674 0.6101706 0.1736349
1zt1 13,ll
13911 [4,11
0.5542636 1.4139378
1.7671637
12.21
[X21
0
1.1975131
t3.21
14321
0.7292301
1.1815807
1.9864396
-...- _-.-..___._.._ (3.4)
____~_.._~.~~.l.~;;.~~~i-.~ ..._........_.^_. 1*.“..“^.^1.^.
” --
(0.1)
0.5127200 1.0115527 0.5048894 I. 1804027
__._ . . .._..
(1.2) __~~_~.._.__.__‘.___.._.__......_..._.....____....__..._..,
L2,0] 13.01 11,Il
I.2733064
__.-.--..-“.~i--
001
0
2
m=O
m=l
m=2__ 4
Foxi@ (au.) Fig. 3.
which the transition occurs, whereas the imaginary part (Im Fnp) (together with the energy splitting AE = E, - Ep) determines the magnitude of d,p. To estimate the range of parameters where the transitions between rotational states of Hz0 are important, let us consider the two most common dependences of the external field on time: (a) linear: Fin. (1) = &or and (b) periodic: Fper.(t) = Fn sin( or). (a) In the first case,
begins at FO = Re F,p, where the channel of transition opens and S,,, ( Fo) --t atin, with increasing amplitude Fo. Eq. (5) gives the amplitude for a single transition. In periodic fields the transitions occur repeatedly. It is easy to obtain the general expression for the probability of multiple transitions induced by the same hidden crossing, P;;‘=;[1-(1-2p,p)“],
J%
Asp = _!_ Im WI
s
[E,(F)
-Es(F)]
dF
= $. 0
Re F,o
(7) where Strn. is a constant which does not depend on o. In Table 1 the quantity &tin, is presented together with related branch points for the states shown in Figs. 1 and 2. (b) In the case of a periodic external field,
(9)
where pan = \A+\* is a probability of a single transition, n is the number of transitions. As follows from if Eq. (9), Pi;’ -+ 0.5 at n + cc monotonically pm@< 0.5 or jumping if pnfi > 0.5. I am grateful to J. Pop-Jordanov for many stimulating discussions and T.P. Grozdanov for the critical reading of manuscript. This work was supported by the Macedonian Ministry of Sciences.
References [ I ] E.D. Giudce, G. Preparate, G. Vitiello Phys. Rev. Lea. 61 =
~per.(Fo) Few
’
where S,,. ( Fo) is a function of the amplitude of oscillation only. Fig. 3 shows S,,. as a function of Fo. As one can see from Fig. 3 and Table 1, each curve
(1988) 1085. [2] T.P. Grozdanov, E.A. Solov’ev, Phys. Rev. A 51 ( 1995) 2630.
[3] [4] [5] [6]
EA. Solov’ev. Sov. Phys. JETP 54 (1981) 893. E.A. Solov’ev, Sov. Phys. Usp. 32 (1989) 228. A. Zwaan, Arch. Neerland IIIA 12 (1929) I. T.P. Grozdanov, M.J. Rakovie. E.A. Solov’ev, Phys. Lett. A 157 (1991) 376.
1998
PHYSICS
ELSEVIER
Physics Letters A 249 (1998)
LETTERS
A
74-76
Hidden crossings and non-adiabatic transitions between rotational states of Hz0 in a slow varying external field E.A. Solov’ev Macedonian
Academy of Sriences and Arts. PO. Box 428, Skopje, Macedonia
Received 20 July 1998; accepted Communicated
for publication by B. Fricke
17 August
1998
Abstract The hidden crossings of rotational energy levels of a Hz0 molecule embedded in an electric field are studied. It is found that non-adiabatic transitions associated with the hidden crossings become significant if the product FOO 2 lo-*, where FO is the amplitude of the periodic field and w is its frequency (both in atomic units). @ 1998 Elsevier Science B.V.
In a simple approximation the transitions between rotational states induced by the interaction of an electric dipole of a molecule H20 with a time-dependent external field are described by the Schrodinger equation t (see, e.g., Ref. [ 1I )
(
-&L’-F(ot).d @(ii,t)
=i
a$$,
at
r>
$<;i,t)
=
cp,<&F(d)) t
x exp
E,( F( ot’) dt’
-i (
s
>
,
(2)
where (~,(a, F) and E,(F) are the eigenfunctions and eigenvalues of the instantaneous Hamiltonian, 9 (1)
>
where F is an external electric field, o is the rate of changing of the electric field, which further is assumed to be a small parameter in the adiabatic approximation, L, I and d are the angular momentum, momentum of inertia and electric dipole vector of the molecule (in the case of HZO: I = 5692 a.u., IdI = 0.756 a.u. [l]), 3 = d/d determines the direction of the dipole vector. In the leading order of the adiabatic approximation the solution of Eq. ( 1) reads
] Atomic units are used throughout
this work.
- $Lz
- Fad
>
(o&F)
=E,(F)po,(&F), (3)
cy is the set of quantum numbers of eigenstates. Since in the limit F -+ 0 the eigenfunctions are transformed into spherical functions, we will specify the adiabatic states by their spherical quantum numbers at F = 0, i.e. ff = llm] . In the frame of reference with the z-axis directed along the external field F, Eq. (3) admits separation of the variables in spherical coordinates 6, 4. Substitution of the wavefunction in the form cpa(d, F) ) = problem V(8) e impleads to the one-dimensional
03759601/98/$ - see. front matter @ 1998 Elsevier Science B.V. All rights reserved. PIISO375-9601(98)00649-S
E.A. Solov’evlPhysics
Letters A 249 (1998)
74-76
15
Re(F)xlO’ (a.u.) Fig. 2.
where 0.5
1.5
1.0
Fx103
2.0
(au)
Fig. 1.
1 d . dV(@ -sm(8)dB sin(O) d6’ +
(
2ZE,(F)
=o.
- 21Fdcos(8)
m2 - ~ V(O) sin’( 0) > (4)
Fig. 1 shows adiabatic energy curves with 1 < 4 and m 6 2. The calculation has been done using a similar code as previously developed for the two-Coulomb center problem [ 2,3]. The adiabatic approximation is based on the fact that the set of energy curves E,(F) of the same symmetry (in our case, having the same value m) are different sheets of a single analytical function E(F); they are connected pairwise (Ia) - I/3)) at certain branch points Fap in the complex F-plane. In this approach 2 , the transition between two adiabatic states ]LY)and I/?) can be obtained as a result of analytical continuation of the wavefunction (2) along the contour Cap (in the complex r or F planes, which are connected analytically) enclosing the related branch point Fmp (Zwaan’s method [ 51) . This procedure gives the following expression for the amplitude of transition, Aap = e - d”p
(5)
‘The adiabatic approach is described in detail, for instance, in review 14)).
The exact crossing of energy surfaces E,(F) and Z+(F) at the branch point F+ reflects the strong coupling of the adiabatic states at the real axis F caused by two different mechanisms: Landau-Zener avoided crossings and hidden crossings. Initially hidden crossings between adiabatic states have been revealed in slow atomic collisions [ 3] . The hidden crossings appear whenever the classical trajectory related to the adiabatic state collapses into an unstable periodic orbit. Then the intensity of the transition is determined by the Lyapunov exponent of this orbit [ 61. The hidden crossings are alternatives to the well-known Landau-Zener avoided crossings, which obey the resonant underbarrier interaction of adiabatic states located in different potential wells. In contrast to the Landau-Zener crossings, the hidden crossings always exist and provide a full description of the non-adiabatic transitions. The characteristic of hidden crossings is that they are not distinguished in the conventional pattern of adiabatic energy curves at a real value of a slow varying parameter and their description requires direct computation in the complex plane of a parameter. Fig. 2 illustrates the distribution of the branch points F,p in the complex F-plane for the set of energy curves in Fig. 1. The complex quantity Fop is of major practical importance. Its real part (Re F,p) corresponds to the strength of the electric field at
76
E.A. Solov’ev/Physics
Letters A 249 (1998) 74-76
Table 1 Branch points F,+q (multiplied by 103) and parameters &in (multiplied by 106)
1:
I4
IP)
Re F,p
Im Fma
&in
[Lull 10.01
[I+ I,tn] Il.01
x10” (a.u.) 0
x IO3 (au) 0.2181081
x IO6 0.0030240
bF
I-
Il.01
I2,Ol
0.3824243
0.5483754
0.1444546
2
[3*01 14.01 12.11
1.0638385
0.9025318
*Qg
2.0355 I97 0
0.6216384
0.3379674 0.6101706 0.1736349
1zt1 13,ll
13911 [4,11
0.5542636 1.4139378
1.7671637
12.21
[X21
0
1.1975131
t3.21
14321
0.7292301
1.1815807
1.9864396
-...- _-.-..___._.._ (3.4)
____~_.._~.~~.l.~;;.~~~i-.~ ..._........_.^_. 1*.“..“^.^1.^.
” --
(0.1)
0.5127200 1.0115527 0.5048894 I. 1804027
__._ . . .._..
(1.2) __~~_~.._.__.__‘.___.._.__......_..._.....____....__..._..,
L2,0] 13.01 11,Il
I.2733064
__.-.--..-“.~i--
001
0
2
m=O
m=l
m=2__ 4
Foxi@ (au.) Fig. 3.
which the transition occurs, whereas the imaginary part (Im Fnp) (together with the energy splitting AE = E, - Ep) determines the magnitude of d,p. To estimate the range of parameters where the transitions between rotational states of Hz0 are important, let us consider the two most common dependences of the external field on time: (a) linear: Fin. (1) = &or and (b) periodic: Fper.(t) = Fn sin( or). (a) In the first case,
begins at FO = Re F,p, where the channel of transition opens and S,,, ( Fo) --t atin, with increasing amplitude Fo. Eq. (5) gives the amplitude for a single transition. In periodic fields the transitions occur repeatedly. It is easy to obtain the general expression for the probability of multiple transitions induced by the same hidden crossing, P;;‘=;[1-(1-2p,p)“],
J%
Asp = _!_ Im WI
s
[E,(F)
-Es(F)]
dF
= $. 0
Re F,o
(7) where Strn. is a constant which does not depend on o. In Table 1 the quantity &tin, is presented together with related branch points for the states shown in Figs. 1 and 2. (b) In the case of a periodic external field,
(9)
where pan = \A+\* is a probability of a single transition, n is the number of transitions. As follows from if Eq. (9), Pi;’ -+ 0.5 at n + cc monotonically pm@< 0.5 or jumping if pnfi > 0.5. I am grateful to J. Pop-Jordanov for many stimulating discussions and T.P. Grozdanov for the critical reading of manuscript. This work was supported by the Macedonian Ministry of Sciences.
References [ I ] E.D. Giudce, G. Preparate, G. Vitiello Phys. Rev. Lea. 61 =
~per.(Fo) Few
’
where S,,. ( Fo) is a function of the amplitude of oscillation only. Fig. 3 shows S,,. as a function of Fo. As one can see from Fig. 3 and Table 1, each curve
(1988) 1085. [2] T.P. Grozdanov, E.A. Solov’ev, Phys. Rev. A 51 ( 1995) 2630.
[3] [4] [5] [6]
EA. Solov’ev. Sov. Phys. JETP 54 (1981) 893. E.A. Solov’ev, Sov. Phys. Usp. 32 (1989) 228. A. Zwaan, Arch. Neerland IIIA 12 (1929) I. T.P. Grozdanov, M.J. Rakovie. E.A. Solov’ev, Phys. Lett. A 157 (1991) 376.