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Vibration–rotation interaction potential for H2O–A system
Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
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Vibration–rotation interaction potential for H2O–A system V.I. Starikov a,b,n a b
Tomsk State University of Control System and Radio Electronics, Tomsk 634050, Russia Tomsk Polytechnic University, Tomsk 634050, Russia
a r t i c l e i n f o
abstract
Article history: Received 11 October 2014 Received in revised form 23 December 2014 Accepted 24 December 2014 Available online 2 January 2015
The contact transformation method is used to derive an effective interaction potential for H2O–A system. The expressions for the second order vibrational and rotational correction terms are obtained. The contributions of the third order, which take into account the Coriolis and Fermi resonances in H2O molecule, are obtained too. The matrix elements of a transformed interaction potential that are necessary for calculating the broadening parameters of H2O lines perturbed by mono-atomic gases are presented. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Perturbation theory Contact transformations Effective interaction potential H2O Interruption functions
1. Introduction Knowledge of the water vapor absorption spectra in the presence of mono-atomic gases is necessary for many applications, mainly for astronomical ones. Water vapor has been detected in many objects of the Solar System, including Venus, Mars, and comets, and was observed in the IR spectra of brown dwarfs, red giant stars, and cold dark molecular clouds. The measured broadening coefficients of H2O molecule in the mixtures with mono-atomic gases Ar, He, Ne, Kr, and Xe are presented in Refs. [1–11]. To calculate the broadening coefficients γ (line-width) and δ (line shift) the interaction potential V(r) (r is the distance between the center of mass of H2O molecule and atom A) as well as rovibrational energies E and wave functions Ψ have to be known. The energies E and functions Ψ must be calculated by solving the Schrodinger equation HΨ ¼EΨ, where H is the Hamiltonian of a vibrating–rotating molecule. In practice the functions Ψ~ instead n Correspondence address: Tomsk State University of Control System and Radio Electronics, Tomsk 634050, Russia. Tel.: þ7 3822 492044; fax: þ7 3822 492086. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.jqsrt.2014.12.018 0022-4073/& 2015 Elsevier Ltd. All rights reserved.
of Ψ are used in calculations of γ and δ. The functions Ψ~ are determined by using the effective Hamiltonian Heff extracted from the transformed Hamiltonian Н~ ¼ U þ HU, where U¼ exp( iS) is the unitary operator of contact transformation and Ψ~ ¼ eiS Ψ . To use the wave functions Ψ~ we have to substitute all matrix elements 〈Ψ|V(r)|Ψ〉 which appear in the calculations of γ and δ for the matrix 0 elements 〈Ψ~ jV~ ðrÞjΨ~ 〉, where V~ ðrÞ ¼ eiS VðrÞe iS is a transformed interaction potential. In Refs. [6,7] the expression for isotropic part, V~ isot , of the transformed interaction potential V~ ðrÞ was obtained. The transformed potential V~ isot depends on the vibrational and rotational operators. This fact explains the unusual rotational dependence of the shift coefficients δ of H2O lines perturbed by He [6–8]. The aim of the present paper is to derive the total expression for the transformed interaction potential V~ ðrÞ for the H2O–A system and the corresponding matrix 0 elements 〈Ψ~ jV~ ðrÞjΨ~ 〉 which are necessary for calculating the broadening coefficients γ and δ. 2. Principle of calculation In the present section only a shortcut presentation is given. The details may be found in Refs. [12,13]. Let V be an
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V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
interaction potential expanded in order of magnitude with respect to the parameter λ as
3. Initial interaction potential
V ¼ V 0 þ λV 1 þ λ2 V 2 þλ3 V 3 þ…
The interaction potential between H2O molecule and atom A is usually taken in the form of atom–atom potential, which is the sum of pair potentials ! 3 X di ei aa V ¼ V 1;2 ¼ 6 ; ð9Þ r 12 ri i i¼1
ð1Þ
It is necessary to perform on the potential V the same set of contact transformations as those used to go from an initial Hamiltonian H to the transformed Hamiltonian Н~ ¼ eiλS He iλS . When performing the unitary transformation, V is replaced by the transformed potential V~ ¼ eiλS Ve iλS ¼ V þ λ½iS; V þ λ2 =2!½iS; ½iS; V þ …
ð2Þ
The transformation operator S is written in the following form: S ¼ S1 þλS2 þ …
ð3Þ
which, in turn, are modeled by the Lennard–Jones (6)–(12) potentials. In formula (9), the index i refers to the i-th atom of the H2O molecule and di and ei are the atomic pair energy parameters. When expressed in terms of rotational matrices Dlmμ (l r2) this potential formally may be written as Vðr; qÞ ¼
2 l X X l ¼ 0 μ ¼ l
In the series of contact transformations the first, second, etc. transformations are V ð1Þ ¼ eiλS1 Ve iλS1 V ð2Þ ¼ eiλ
2
S2
V ð1Þ e iλ
2
S2
ð4Þ
The transformed potential (2) then is written as ðkÞ ðkÞ V~ ¼ V~ 0 þ V~ 1 þ V~ 2 þ… ¼ V ðkÞ 0 þ V 1 þV 2 þ::
ð5Þ
The upper and lower subscripts represent the number of used transformations and the order of λ, respectively. It is shown in Refs. [6–8] that for an H2O–He system it is necessary to use some terms from a transformed isotropic potential V~ isot , obtained in the third order of perturbation theory. For any commutator [iS, V] in Eq. (2) the relation ½iS; V ¼ ½iS; VV þ λ½iS; V R
ð6Þ
[12,13] is used. Here the subscripts V and R denote the vibrational or rotational commutator, respectively. We assume that the zero-order term V0 does not depend on the vibrational variable and [iS, V0]V ¼ 0. In this case two transformations are required to obtain V~ up to λ3. The contributions into twice-transformed up to λ3 potential (5) take the form
V~ 2 ¼ V 2ð1Þ ¼ V 2 þ ½iS1 ; V 1 V þ ½iS01 ; V 0 R V~ 3 ¼ V 2ð2Þ ¼ V 3 þ ½iS1 ; V ″2 V þ ½iS01 ; V 1 R þ½iS2 ; V 1 V þ ½iS2 ; V 0 R ; ð7Þ where ð8Þ
and S10 ¼S1 þSR. The operator SR was used in Refs. [14,15] to obtain a reduced Hamiltonian for asymmetric top molecules. The formulas for the first and second order transformation operators S0 1 and S2 are presented in Refs. [12,13] and they are presented in Appendix A for convenience. The commutation rules for some vibrational and rotational operators are given in Appendix B.
ð10Þ
Here we assume that the interaction potential (10) depends on the vibrational dimensionless normal coordinates qi(i¼ 1,2,3) of the H2O molecule. The index m¼0 on matrix Dlmμ is omitted. The expression (10) is taken from Ref. [17] where it is written for the functions Vlμ(r,q ¼0). Introducing q – dependence of these functions – we can write ( ) jμj l jμj l 4 X jμj l D12 þ t ðqÞ jμj l E 6 þ t ðqÞ V lμ ðr; qÞ ¼ b c ð11Þ 0 t 0 t r 12 þ t r6 þ t t¼0 The expressions for the quantities ð:::Þ Dð:::Þ ð:::Þ ðq ¼ 0Þ and 0Þ as well as the coefficients b and c are given in Ref. [17]. The quantities D12 þ t, E6 þ t (other indices are omitted) depend on the energy parameters eH eH–A, eO eO–A, dH dH–A, dO dO–A and on the molecular parameters rH1, rH2, rO and 2β which are the H1–G, H2–G, O–G distances and the angle H1GH2, where G is the center of mass of the H2O molecule. These molecular parameters correspond to the instantaneous configuration of a molecule and may be expanded in the normal coordinates qi, for example, as follows: ð:::Þ ð:::Þ Eð:::Þ ðq ¼
r H1 ðqÞ ¼ r H þ
X
r H1;i qi þ
i
V~ 0 ¼ V 0 V~ 1 ¼ V 1
1 V ″2 ¼ V 2 þ ½iS1 ; V 1 V 2
V lμ ðr; qÞDlμ
1X r q q þ… 2 i;j H1;ij i j
ð12Þ
ð:::Þ ð:::Þ In Ref. [17] the quantities ð:::Þ Dð:::Þ Eð:::Þ ðq ¼ 0Þ ð:::Þ ðq ¼ 0Þ and are determined for the equilibrium configuration of the H2O molecule in which the molecular parameters (rH1 ¼ rH2 ¼rH, rO and β) do not depend on qi. The expansions of the molecular parameters rH1, rH2, rO and β in the coordinates qi lead to the expansions of the quantities D12 þ t and E6 þ t in qi
D12 þ t ðqÞ ¼ D12 þ t ð0Þ þ
X
D12 þ t; i qi þ
i
1X D q q þ… 2 i;j 12 þ t; ij i j ð13Þ
and of the functions V lμ ðr; qÞ ¼ V lμ ðrÞ þ
X i
V lμ;i ðrÞqi þ
1X l V ðrÞqi qj þ … 2 i;j μ;ij
ð14Þ
V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
When expansion (14) is substituted, the interaction potential (10) takes the form (1) with X V0 ¼ V l ðrÞDlμ l;μ μ o X nX V1 ¼ V l ðrÞDlμ qi l;μ μ;i i
o 1X nX V2 ¼ V l ðrÞDlμ qi qj l;μ μ;ij 2 i;j
ð15Þ
parameters R V l;α are defined as μ;k x y z
l ¼ 1; μ ¼ 1 pffiffiffi 1 ði= 2ÞSxx k V0 pffiffiffi yy 1 ð1= 2ÞSk V 0 pffiffiffi 1 i 2Sxz 3 V0
l ¼ 2; μ ¼ 2 2 2iSxz 3 V2
2 2iSzz k V2
l ¼ 2; μ ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 iSxx 3=2 V 20 Þ k ðV 2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffi Syy ðV 22 3=2 V 20 Þ k qffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 iSxz 3=2 V 20 Þ 3 ðV 2 þ ð20Þ
In Sections 4–8 the dependence of the potential (10) on the distance r is omitted for brevity.
These parameters satisfy the relations R
4. Once transformed interaction potential (up to λ2) To obtain V~ ¼ V 0 þ V 1 þ V~ 2 it is necessary to calculate two commutators, [iS1,V1]V and [iS0 1,V0]R. The required calculation commutators are given in Appendix B. The substitution S0 1, V0, and V1 in Eq. (7) for V~ 2 results X ij l X l;ij Dl ð V~ μ pi pj þ V~ μ qi qj Þ V~ 2 ¼ l;μ μ i;j X l;α 1X 1X l;αβ V~ μ;k pk þ fDl ; J g fDl ; J J gV~ þ 2 l;μ;α μ α k 2 l;μ;α;β μ α β μ ð16Þ Here {A,B}¼AB þBA is an anticommutator. Other quantities are defined as a sum of two terms ij
51
l V~ μ ¼ ij V lμ þ ij V l;R μ ;
V 1;x 1;k R 2;x V 1;k R 2;z V 1;3
R 1;z ¼ R V 1;x ; R V 1;y ¼ R V 1;y ; R V 1;z 1;3 ¼ V 1;3 ; 1;k 1;k 1;k R 2;x R 2;y R 2;y ¼ R V 2;x ; R V 2;x 2;3 ¼ V 2;3 ; V 1;k ¼ V 1;k ; 1;k R 2;z R 2;z ¼ R V 2;z 1;3 ; V 2;k ¼ V 2;k ;
where k ¼1 or 2. The non-zero parameters R V l;αβ are defined as μ;k pffiffiffi R 1;xz R 1;xz 1 V 1 ¼ V 1 ¼ s111 V 0 = 2; pffiffiffi R 1;yz 1 V 1 ¼ R V 1;xz 1 ¼ is111 V 0 = 2; qffiffiffiffiffiffiffiffiffiffiffiffi ffi R 2;yz 2 V 1 ¼ R V 2;yz 3=2 V 20 Þ; 1 ¼ is111 ðV 2 þ qffiffiffiffiffiffiffiffiffiffiffiffi ffi R 2;xz 2 V 1 ¼ R V 2;xz 3=2 V 20 Þ; 1 ¼ s111 ðV 2 R
V 2;xy ¼ R V 2;xy ¼ 2is111 V 22 2 2
ð21Þ
Formally, all obtained expressions (16)–(21) with l¼ 1 coincide with the corresponding expressions for the P transformed dipole moment operator μ~ Z ¼ φα μ~ α from
l;ij R l; V~ μ ¼ V l;ij μ þ V μ;ij ; l;α þ R V l;α ; V~ μ;k ¼ V l;α μ;k μ;k
α
ð17Þ
Ref. [13] if φα are defined by Eq. (B4) and V10 ¼μz, pffiffiffi V 17 1 ¼ ð 8 μx iμy Þ= 2.
obtained from the vibrational and rotational commutators, respectively. The terms from the vibrational commutator are X ijk l ij l Vμ ¼ 3S V μ;k ;
5. Accidental resonances in H2O molecule and contributions of the third order in the twice-transformed interaction potential
l;αβ R l;αβ V~ μ ¼ V l;αβ μ þ Vμ ;
k
X k l 1 l Sij V μ;k ; V l;ij μ ¼ V μ;ij þ 2 k X α kn l ¼ S V μ;n ; V l;α μ;k n
V l;αβ μ ¼
X
Sαβ V lμ;k k
ð18Þ
k
and The first two terms ij V l;R μ rotational commutator [iS0 1,V0]R are
R
V lμ;ij (17) from the
1y R l S V ; 2 ij μ 1 2Þ R V lμ;ij ¼ y Sij R V lμ ; 2 1Þ ij V l;R μ ¼
where R
V lμ ¼
1 pffiffiffiV 10 δl;1 þ V 22 2
ð19Þ rffiffiffi ! ! 3 2 δμ; 7 1 μ V δ 2 0 l;2
and δl,m is a delta symbol. We take into account that among the coefficients αSij and αSij only yS13 ¼ yS31 a0 and y S23 ¼ yS32 a0 (see Appendix A). The compact expressions for the terms R V l;α and R V l;αβ (17) from the rotational μ;k μ;k commutator [iS0 1,V0]R are not obtained. The non-zero
The rovibrational energies E and wave functions Ψ~ of H2O molecule are usually obtained using the effective Hamiltonian Heff extracted from the transformed Hamiltonian Н~ . The effective Hamiltonian Heff is build for a given polyad (group) of interacting vibrational states. These polyads are formed according to the rule 2v1 þ2v3 þ v2 ¼ p ¼0, 1, 2, 3…., where v1, v3, and v2 are the vibrational quantum numbers. This rule is based on the relation ω1 Eω3 E 2ω2 and it generates the series of accidental resonances in the H2O molecule which have to be taken into account in the transformed isotropic potential [6–8]. In the present study our efforts are concentrated on the Cor Fer contributions V~ 3 and V~ 3 which describe the Coriolis interactions (ω1 Eω3) between vibrational states |2〉¼| v1 þ 1v2v3〉 and |3〉¼|v1v2v3 þ1〉 and Fermi interactions (ω1 E2ω2) between vibrational states |1〉¼|v1v2 þ2v3〉 and |2〉¼|v1 þ1v2v3〉, respectively. These contributions appear in the twice-transformed third-order interaction potential V~ 3 from Eq. (7). The Cor general expression for V~ 3 is X Cor 1X 1 V~ 3 ¼ fDl ; J g i;j α Rlμ;ij ðqi pj þ pj qi Þ 2 l;μ;α μ α 2
ð22Þ
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V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
matrix elements 〈njV~ jm〉 in which V~ ¼ V 0 þV 1 þ V~ 2 and V0, V1 and V~ 2 are defined by Eqs. (15) and (16), respectively.
where X j l;α 1X α ″ l;jk α jk l α l Rμ;ij ¼ ð Sik V μ Sik V μ Þ 2 Sik V μ;k 2 k k X α kj l R l;α þ2 Si V μ;k þ R V l;α μ;ij þ ð2Þ V μ;ij
ð23Þ
k
and
l,ij l,ij Vμ ¼Vlμ,ij/2 þVμ .
Formulas for R V l;α μ;ij are obtained from
the formulas for R V l;α (20) in which μ;k by
αβ j Si.
Analogously Vlμ,i
R l;α ð2Þ V μ;ij
αβ Sk
must be substituted
are obtained from
Vlμ.
R
V l;α μ;i by
for substituting Taking into account only the isotropic part of interaction potential (l¼0 in Eq. (22)) we can obtain that for H2O molecule Cor V~ 3 ¼ J y ðy R13 q1 p3 þ y R31 q3 p1 Þ;
ð24Þ
0;y R where y Rij ¼ y R00;ij . For this quantity R V 0;y 0;13 ¼ ð2Þ V 0;13 ¼ 0 in Fer (23). The Fermi contribution V~ 3 is extracted from the general expression for X l X Fer 1 ~ l;ij l ~ V~ 3 ¼ V V D q q q þ ðq p p þp p q Þ i j m m i j i j m μ;ijm μ;m μ l;μ i;j;m 2 ð25Þ
6.1. Effective interaction potential in a given vibrational state The effective interaction potential (up to λ2) in the vibrational state |n〉 ¼|v1v2v3〉 is obtained as X X ðnÞ 1 ii ~ l l l ð Vμ D fV þ v þ V~ ¼ 〈njV 0 þ V 1 þ V~ 2 jn〉 ¼ i μ l;μ μ 2 i l;ii 1 þ V~ μ Þg þ 2
ðnÞ V~ ¼
X
l l;12 l;22 Dl ½3V~ μ;122 q1 q22 þ V~ μ;2 ðq2 p1 p2 þp1 p2 q2 Þ þ V~ μ;1 q1 p22 l;μ μ
l 1 V~ μ;ijm ¼ V lμ;ijm þ 2 6 l;ij V~ μ;m ¼
X n
Snmi '' V l;nj μ þ
X n
Sijm V lμ;n þ
1y S V l RV l ; 2 ij μ;m μ
X ijn l;nm 1y l ð6S V μ 4Sjmn in V lμ þ3Sijn S V l RV l m V μ;n Þ þ 4 ij μ;m μ n ð27Þ
6. Vibrational matrix elements of the transformed up to the second-order interaction potential To calculate a line width γif or line shift δif for the rovibrational transition (i)-(f) the matrix elements 〈Ψ~ f jV~ jΨ~ f 〉 and 〈Ψ~ i jV~ jΨ~ i 〉 have to be evaluated. For the functions Ψ~ f and Ψ~ i we can write [13] P P J; K; γ〉 jΨ~ f 〉 ¼ Ψ ðv1 v2 v3 JK a K c Þ〉 ¼ m〉 C ðmÞ k m
l;μ
Dlμ V~ μ ðnÞ þ
l V~ μ ðnÞ ¼ V lμ þ
l
1X l;αβ fDl ; J J gV~ ; 2 l;μ;α;β μ α β μ
X 1 ðvi þ ÞΔV lμ;i ; 2 i
X ϕiik V μ;k 1 V lμ;ii ¼ 2 ωk k l
ΔV lμ;i
ð26Þ In Eq. (26)
l;αβ fDlμ ; J α J β gV~ μ
ð28Þ
k
ð31Þ
ð32Þ
! ð33Þ
and фijk are the cubic anharmonic constants [18] from the intramolecular potential of H2O molecule. The coefficient l l of Dμ in Eq. (31), i.e. V~ μ , may be called the effective vibrational (rotation-free) interaction potential in the state |n〉¼|v1v2v3〉 of H2O molecule. Eq. (32) gives the vibrational correction to the initial potential. It involves the first as well as the second derivatives of the potential with respect to the normal coordinates. The second term in Eq. (31) determines the rotational correction to the effective potenðnÞ tial. It is useful to present V~ (31) in the form ðnÞ ðnÞ ðnÞ V~ ¼ V~ diag þ V~ n:diag
ð34Þ
ðnÞ 〈J,K,γ|V~ n:diag |J,K,γ〉 ¼0.
l
Using approximation 1/2{Dμ, in which l JαJβ}-DμJα Jβ we can obtain a simple equation X ðnÞ ðnÞ l V~ diag ¼ V~ ef f ¼ Dl V~ ðnÞ; ð35Þ l;μ μ μ;ef f where l l l l l l l V~ μ;ef f ðnÞ ¼ V~ μ ðnÞþ V~ μ;rot ¼ V~ μ ðnÞ þ ðJÞ V~ μ U J2 þ ðKÞ V~ μ J 2z þ ðxyÞ V~ μ ðJ 2x J 2y Þ
k
X ð0Þ C k J; K; γ〉 jΨ~ i 〉 ¼ Ψ ð000JK a K c Þ〉 ¼ 0〉
ð30Þ
where
Fer
X
l;μ;α;β
R l Because ii V l;R μ ¼ V μ;ii ¼ 0 in Eq. (17) (see Eq. (19) and Appendix A) then
by the condition 〈1|V~ 3 |2〉 a0. This gives Fer V~ 3 ¼
X
ð36Þ and l 1 1X xx l;yy V~ μ ¼ ðV l;xx ðS þ Syy ÞV lμ;k ; μ þVμ Þ ¼ k 2 2 k k
where the symbol B stands for all the vibrational states of the polyad to which the vibration–rotation state under study belongs and |m〉¼|v1v2v3〉, |0〉 ¼|v1 ¼0 v2 ¼0 v3 ¼0〉. The functions |JKγ〉(γ¼ 71) are defined as pffiffiffi jJKg〉 ¼ fjJK〉 þ γjJ K〉g= 2 jJK ¼ 0; γ〉 ¼ jJK ¼ 0〉; ð29Þ
ðJÞ
where |JK〉 are the symmetric top rotational wave functions. A matrix element 〈Ψ~ f jV~ jΨ~ f 〉 involves the vibrational
ðxyÞ
ðKÞ
l l;zz l V~ μ ¼ V~ μ ðJÞ V~ μ ¼
X
l ðJÞ ~ Szz k V μ;k V μ ; l
ð37Þ
ð38Þ
k
l 1 l;xx l;yy 1X xx l V~ μ ¼ ðV~ μ V~ μ Þ ¼ ðS Sxx k ÞV μ;k 2 2 k k
ð39Þ
V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
The main contribution into the line shift coefficient δif is determined by the first-order interruption function Z
1 1 h ~ ~ ~ S1 ¼ dt 〈Ψ f V isot Ψ f 〉 〈Ψ~ i V~ isot Ψ~ i 〉 ; ð40Þ ℏ 1 which depends on the difference between the isotropic 0 potential V~ isot ¼ V~ 0 in the initial (i) and final (f) rovibrational states. According to Eq. (36) this difference may be presented as 〈Ψ~ f jV~ isot jΨ~ f 〉 〈Ψ~ i jV~ isot jΨ~ i 〉 ¼ ΔV~ vib þ ΔV~ rot ; where ΔV~ vib ¼
ð41Þ
53
involved in Coriolis resonances. In this case Cor Cor Cor 〈Ψ~ f jV~ 3 jΨ~ f 〉 ¼ 〈Ψ~ f 0 jV~ 3 jΨ~ f 0 〉 ¼ R31 f ðv1 ; v3 Þf ðJ; KÞ
where
ð48Þ
R31 ¼1/2{Ry31–Ry13}
and 1=2 f ðv1 ; v3 Þ ¼ ðv1 þ1Þðv3 þ1Þ f
Cor
ðJ; KÞ ¼
X k;k'
0 0 C ð3Þ C ð3Þ C kð2Þ fC ð2Þ 0 g〈JKγ ðiJ y Þ JK γ 〉 k k k0
ð49Þ
In the same way we can obtain the matrix element Fer Fer 0;Ferm 〈Ψ~ f jV~ 3 jΨ~ f 〉 ¼ 〈Ψ~ f 0 jV~ 3 jΨ~ f 0 〉 ¼ 2F 00 f ðv1 ; v2 Þf ðJ; KÞ
ð50Þ
X
v ΔV 00;i ; i ¼ 1;2;3 i
ð42Þ
and X ðmÞ f ðC k Þ2 ðC ð0Þ Þ2 gK 2 k
0X
ΔV~ rot ¼ ðJÞ V~ 0 ½J f ðJ f þ 1Þ J i ðJ i þ 1Þþ ðKÞ V~ 0 0
k
m
ð43Þ (the contribution from the last term of Eq. (36) is omitted). For example, for the ν1 band of H2O molecule in Eq. (42) v1 ¼1, v2 ¼v3 ¼ 0 and ΔV~ vib ¼ ΔV 00;1 . In the approximation of symmetric top molecule Eq. (43) becomes ΔV~ rot ¼ ðJÞ V~ 0 f ðJÞ þ ðKÞ V~ 0 f ðKÞ: 0
0
1=2 f ðv1 ; v2 Þ ¼ ðv2 þ2Þðv2 þ1Þðv3 þ 1Þ X 0;Ferm f ðJ; KÞ ¼ C kð1Þ C ð2Þ k
f ðKÞ ¼ ½K 2f K 2i
ð45Þ
6.2. Off-diagonal vibrational matrix elements Because there are the resonance interactions in the H2O molecule the matrix elements 〈1|V~ |2〉 and 〈2|V~ |3〉 have to be calculated. For the potential V~ 2 (16) the matrix elements 〈1|V~ 2 |2〉 ¼〈v1v2 þ2v3|V~ 2 |v1 þ1v2v32〉¼0 and formally, according to Eq. (16) 〈2jV~ 2 j3〉 ¼ 〈v1 þ1v2 v3 jV~ 2 jv1 v2 v3 þ 1〉 v1 þ1 1=2 v3 þ 1 1=2 X l ¼ Dl V~ l;μ μ μ;1 3 2 2
Fer The term V~ 3 describes the resonance interactions between rovibrational states (f) and (f0 ) with ΔK ¼Kf0 – Kf0 ¼0. Usually they appear for the highly excited rotational states of H2O molecule with K EJ, but the broadening coefficients γ and δ usually are measured for small or middle K. These rotational states may be involved in Fermi resonances with ΔK ¼ 72, 74,… To describe these reso2 2 4 4 nances the rotational operators (J þ þJ ), (J þ þJ ),… have to be introduced into transformed potential. The rotational dependence of Fermi resonance appears in the highly ordered terms of V~ . In the present study the dependence 2 2 of V~ on the operator (J þ þJ ) is introduced in the matrix Fer element 〈1|V~ |2〉 as
3
Fer 〈1jV~ 3 j2〉 ¼ f ðv1 ; v2 ÞfF ð0Þ þF ð2Þ ðJ 2þ þ J 2 Þg
ð46Þ
In this case Fer 0;Fer 2;Fer 〈Ψ~ f jV~ 3 jΨ~ f 〉 ¼ f ðv1 ; v2 Þ½2F ð0Þ f ðJ; KÞ þF ð2Þ f ðJ; KÞ
where X
ϕ13k V lμ;k
k
ðω3 ω1 Þ2 ω2k
ð52Þ
k
ð44Þ
in which K Ka and f ðJÞ ¼ ½J f ðJ f þ 1Þ J i J i þ1 ;
l l V~ μ;1 3 ¼ V~ μ;13 þ
for two states (f) ¼(v1v2 þ2v3)[Jf Kaf Kcf ] and (f0 )¼ (v1 þ1v2v3) [Jf0 Kaf0 Kcf0 ] involved in Fermi resonances. In Eq. (50) only the isotropic part of transformed potential is taken into account,
1 3 0 1 0;22 0;12 F 00 F ð0Þ ¼ pffiffiffi V~ 0;122 V~ 0;1 þ V~ 0;2 ; ð51Þ 2 2 2
þ ðy S13 þ y S13 ÞR V lμ
ð47Þ
where f
2;Fer
ðJ; KÞ ¼
X k
For H2O molecule this term is not equal to zero for l¼1, μ¼ 71. In the initial potential (10) the term with V lμ¼¼17 1 ðr; q ¼ 0Þ is absent and it will be omitted in subsequent calculations. 7. Rovibrational matrix elements of resonance terms Cor Fer The resonance terms V~ 3 and V~ 3 are defined by Eqs. (24) and (26), respectively. Let for the two transitions (i)(f) and (i0 )-(f0 ) the upper rovibrational states (f)¼ (v1 þ1v2v3)[JfKafKcf] and (f0 )¼(v1v2v3 þ1)[Jf0 Kaf0 Kcf0 ] be
ð53Þ
ðC ð1Þ C kð2Þ7 2 þ C kð2Þ C kð1Þ7 2 Þf½JðJ þ 1Þ k
KðK 7 1ÞJðJ þ 1Þ ðK 7 1ÞðK 72Þg1=2
ð54Þ
8. Interruption functions To calculate the line broadening coefficients, γ and δ, the interruption functions of the first, S1(rc,v), and second, S2(rc,v), order have to be known [17,19,21] (rc is the closest approaching distance between colliding partners and v is their relative velocity). The relations between the function ð:::Þ ð:::Þ S2(rc,v) and the quantities ð:::Þ Dð:::Þ Eð:::Þ ðq ¼ 0Þ of ð:::Þ ðq ¼ 0Þ and
54
V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
the initial interaction potential V(r,q¼0) (10) and (11) are established in Refs. [17,21]. With the little modifications they may be adopted to the effective potential (35). According to Eqs. (11), (14) and (15) the functions Vlμ(r), Vlμ,i(r), and Vlμ,ij(r) may be presented as ! þt þt 4 X V l;6 V l;12 μ μ 6 þ t þ 12 þ t ; V lμ ðrÞ ¼ r r t¼0 ! l;6 þ t þt 4 X V μ;i V l;12 μ;i V lμ;i ðrÞ ¼ 6 þ t þ 12 þ t ; r r t¼0 ! l;6 þ t þt 4 X V μ;ij V l;12 μ;ij l 6 þ t þ 12 þ t ð55Þ V μ;ij ðrÞ ¼ r r t¼0 When these functions are substituted, the effective potenðnÞ tial V~ ef f (35) becomes 0 1 ðnÞ ~ l;6 þ t ðnÞ ~ l;12 þ t 4 X X Vμ Vμ ðnÞ l @ V~ ef f ðrÞ ¼ ð56Þ D þ 12 þ t A; l;μ μ r6 þ t r t¼0 where, according to (36), ðnÞ
in expression (33) for ΔV~ vib , in expressions (37)–(39) Cor for ΔV~ rot , in formulas (17) and (23) for ΔCp , and in Fer formula (51) for ΔCp all derivatives Vlμ(r),Vlμ,i(r), and Vlμ, l,p l,p l,p ij(r) must be substituted by the derivatives Vμ ,Vμ,i, and Vμ,ij, respectively. Formula (59) may be complicated in an application. Usually the expansion (11) (with l¼ 0) for the isotropic potential is modeled by the function V mod isot ¼
C 6 C 12 þ ; r 6 r 12
in which C6 ¼4εσ6, and C12 ¼ 4εσ12, and ε and σ are known parameters. In Refs. [6–8] a simplified form of S1 which mod corresponds to Visot (62), was used in the calculation of H2O line shift in the H2O–He system. This form depends on some variable parameters. The simplification and parameterization of S1 (59) may be performed by placing in (59) Cor Fer ΔC8 ¼ΔC10 ¼ΔC14 ¼ΔC16 ¼ΔC12 ¼ ΔC12 ¼0 so that " ! # 3π ΔC rot ΔC Cor ΔC Fer 21 ΔC vib vib 6 6 6 12 þ a12 S1 ¼ ΔC 6 a6 1 þ þ þ 6 8vℏr 5c 32 ΔC vib ΔC vib ΔC vib ΔC vib 6 6 6 6 rc
ð63Þ
l;p l;p l;p l;p l;p l;p l;p V~ μ ¼ V~ μ ðnÞ þ V~ μ;rot ðnÞ ¼ V~ μ ðnÞ þ ðJÞ V~ μ J2 þ ðKÞ V~ μ J 2z þ ðxyÞ V~ μ ðJ 2x J 2y Þ
ð57Þ l;p V~ μ ðnÞ,
l;p …ðxyÞ V~ μ
and p¼6 þt or 12 þt. The expressions for are given by (32), (33), (37)–(39) in which the quantities Vlμ(r),Vlμ,i(r), and Vlμ,ij(r) must be substituted by the quantities Vl,pμ, Vl,pμ,i, and Vl,pμ,ij, respectively. The isotropic potential (l ¼0 in Eq. (56)) may be rewritten in the form ! ðnÞ ðnÞ X C~ 6 þ t C~ 12 þ t ðnÞ þ ð58Þ V~ isot ðrÞ ¼ t ¼ 0;2;4 r 6 þ t r 12 þ t ðnÞ 0;p with C~ p ¼ ðnÞ V~ 0 . The interruption function of the first order S1(rc,v) is defined by Eq. (40). The substitution of Eq. (58) in Eq. (40) results in the following: 3π 5 ΔC 8 35 ΔC 10 21 ΔC 12 a6 ΔC 6 a8 a10 þa12 S1 ¼ 5 6 r 2c 48 r 4c 32 r 6c 8vℏr c 77 ΔC 14 143 ΔC 16 þ a14 þ a16 ð59Þ 128 r 8c 256 r 10 c
The coefficients ap are defined in the way that a6 ¼(8/ 3π)A6,… a12 ¼(8/3π) (32/21) A12, … a16 ¼(8/3π) (256/143) A16 and Z 1 dy Ap ¼ ð60Þ pffiffiffi yp=2 fy 1 þ V isot ðr c Þ yV isot ð yr c Þg1=2 1 In the calculation of integral (40) the kinematical model of Ref. [19] (taking into account the close collisions) and the exact trajectories [20] for the colliding particles are used. In the straight line approximation for the trajectory Visot(rc)¼0 in Eq. (60) and all coefficients ap ¼1 in (59). Each contribution ΔCp from Eq. (59) is written as rot Cor Fer ΔC p ¼ ΔC vib p þ ΔC p þΔC p þΔC p vib ΔCp
ð61Þ
rot ΔCp
and are defined by expresHere the quantities sions (42) and (43) for ΔV~ vib and ΔV~ rot , respectively, Cor Fer Cor Fer ΔCp ¼ 〈Ψ~ f jV~ jΨ~ f 〉 (48) and ΔCp ¼〈Ψ~ f jV~ jΨ~ f 〉 (53). But 3
3
ð62Þ
The often-used expression C6 ¼[μ2 þ3/2uα]α2 relates C6 to the molecular parameter. Here u¼u1u2/(u1 þu2), μ and α are the dipole moment and the averaged polarizability of H2O molecule, α2 is the polarizability of an atom A and u1 and u2 are the ionization energies of the molecule and vib atom, respectively. This expression allows us to write ΔC6 as 2 2 ΔC vib 6 ¼ ½μðf Þ μðiÞ þ3=2uðαðf Þ αðiÞÞα2 ;
ð64Þ
where μ(f), μ(i) and α(f), α(i) are the dipole moments and the averaged polarizability of H2O molecule in the final (v1v2v3) and initial (000) vibrational states, respectively. 0;6 The variable parameters may be introduced as xðJÞ ¼ ðJÞ V~ = ΔC vib 6 ,
6 6 vib vib ðKÞ vib ðKÞ ~ 0;6 6 x6 ¼ V 6 =ΔC 6 and y¼ΔC12 /(σ ΔC6 ). In the Cor vib 6 ðJ; KÞ=ΔC vib the function ΔC Cor 6 =ΔC 6 ¼ R31 f ðv1; v3 Þf 6 Cor
ratio fCor(J,K) may be approximated by the function f (J,K)¼ Cor Cor Cor C1 F(Jf,Kf) C% , in which C1 is some constant, F(Jf,Kf)¼ Cor 1/2 {Jf(Jf þ1) Kf (Kf þ1)} and C% determine the mixing of the final states involved in Coriolis resonances. With a Cor
vib 6 variable parameter r 631 ¼ C Cor the ratio ΔC6 / 1 R31 =ΔC 6 vib ΔC6 becomes vib Cor 6 ΔC Cor 6 =ΔC 6 ¼ r 31 f ðv1 ; v3 ÞFðJ f ; K f ÞC %
ð65Þ
In the same way in (52)–(54) we can approximate f0, Fe 0,Fer Fer ,2,Ferm Ferm (J,K)¼C0 C% , f2,Fer(J,K)¼C2 C% F2(Jf,Kf), where C0 Ferm 0,Fer 2,Fer and C2 are some constants and C% and C% determine the mixing of the final states, involved in Fermi resonances with ΔKf ¼0, 72, respectively, F2(Jf,Kf) ¼F(Jf, Kf) F(Jf, Kf þ1). The introduction of the variable parameters by the expressions
Fer
0;6Þ vib ð0Þ ¼ 2C Fer r ð12 0 F =ΔC 6 ; 2;6Þ vib ð2Þ ¼ C Fer r ð12 2 F =ΔC 6
V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
55
allows us to rewrite Eq. (63) in the form
Appendix A. Transformation operators
n 3π ðKÞ Cor Cor S1 ¼ ΔC vib a6 ð1 þ xðJÞ 6 6 f ðJÞ þ x6 f ðKÞ þ r 31 f ðv1 ; v3 ÞFðJ f ; K f ÞC % 8vℏr 5c
S01
h i 21 σ 6 ð2;6Þ 0;Fer 2;Fer f ðv C þr F ðJ ; K ÞC ; v Þ þ a y þ ðrð0;6Þ 2 1 2 12 f f % % 12 12 32 r c
ð66Þ To modify the relations [17,21] between the interruption functions S2(rc,v) and the potential parameters it is useful to rewrite an effective potential (56) in the form 0 1 jμj ~ l jμj ~ l 4 X X ðnÞ jμj l D 12 þ t A jμj l E 6 þ t l ~ @ ; ð67Þ V ef f ðrÞ ¼ D 0 ct 6 þ t þ 0 bt 12 þ t l;μ μ r r t¼0 in which jμj ~ l D 12 þ t
l;12 þ t 2 l;12 þ t 2 J2 þ ðKÞ V~ μ J z þ ðxyÞ V~ μ ðJ x J 2y Þ
o
ð68Þ
This formula determines the vibrational and rotational ~ ð:::Þ and ð:::Þ E~ ð:::Þ . In the dependence of the quantities ð:::Þ D ð:::Þ ð:::Þ same way this formula determines the vibrational and rotational dependence of the interruption functions S2 involved in the calculations of the broadening coefficients γ and δ. In formulas for the functions S2(rc,v) [17,21] the ~ ð:::Þ and ð:::Þ E~ ð:::Þ have to be used instead of quantities ð:::Þ D ð:::Þ
Dð:::Þ ð:::Þ
and
ð:::Þ ð:::Þ ð:::Þ Eð:::Þ .
¼ S1 þSR ¼ S30 þ S21 þS12 þ SR
ðA1Þ
The first and second indices in S1 show the degree of the vibrational (qk or pk ¼ i∂/∂qk) and rotational (Jα) operators, respectively; qk are the dimensionless normal coordinates of H2O molecule (k ¼1, 2, 3) and Jα(α¼ x, y, z) represent the angular momentum components of the H2O molecule in the molecular axis system. Here [12,13] X kij 1 S pk pi pj þ Sjki ðqk qi pj þpj qk qi Þ S30 ¼ k;i;j 2 X αβ S12 ¼ S J J p α;β;k k α β k S21 ¼
( X 1 jμj l jμj l þt 1 ¼ 0 D12 þ t þ ð0 b12 þ t Þ vi þ ΔV l;12 μ;i 2 i l;12 þ t þ ðJÞ V~ μ
The first-order transformation operator S10 is written as
1X J ðα S q q þ α Skl pk pl Þ 2 k;l;α α kl k l
SR ¼ s111 ðJ x J y J z þJ z J y J x Þ ¼
ðA2Þ
i is111 h 2 J þ ðJ z þ 1Þ ðJ z þ 1ÞJ 2 2
and Sαβ ¼ ð Bαβ =ωk Þ; k k Skij ¼ фkij ωk ωi ωj =ð3Ωkij Þ; Sjki ¼ фkij ωj ðω2j ω2k ω2i Þ=ð2Ωkij Þ; α α
Skl ¼ 4Bα ξαkl ðωk ωl Þ1=2 =ðω2k ω2l Þ; Skl ¼
α
Skk ¼ 0;
2Bα ξαkl ðω2k þ ω2l Þ=½ðωk ωl Þ1=2 ðω2l ω2l Þ;
The analysis of measured broadening coefficients of H2O lines [6–8,11], perturbed by He, revealed some new details. Firstly it was shown that the shift coefficients δ depend on the sign of ΔKa ¼Kaf–Kai in the transition (i)(f), (i) (000) [JiKaiKci], (f) (v1v2v3) [JfKafKcf]. Secondly, the calculated line shifts coefficients δ for two transitions (i)-(f) and (i0 )-(f0 ) depend on the existence of the resonance interactions between upper rovibrational states (f) and (f0 ). These details were explained by the rotational dependence of the first-order interruption function S1(rc,v). It is shown in the present paper that the second-order interruption functions S2(rc,v) used in the calculations of the broadening coefficients γ and δ are vibrational and rotational dependent too. This dependence is connected with the rovibrational dependence of an effective interaction potential (which is generated by the vibrational dependence of the intra-molecular parameters rH, rO, and β of H2O molecule) and with the use of the effective wave functions. The application of the obtained expressions for the effective interaction potential of the concrete H2O–A system will be discussed in the next paper.
α
Skk ¼ 0;
Ωkij ¼ ðωk þ ωj þωi Þð ωk þ ωj þωi Þðωk ωj þ ωi Þðωk þ ωj ωi Þ ðA4Þ
ð:::Þ
9. Conclusion
ðA3Þ
αβ
In these equations Bα and Bk are the rotational constants and rotational derivatives, respectively, ωk are the frequencies of the harmonic vibrations. Modifications of the formulas (A4) in the case of Coriolis and Fermi resonances are discussed in [13]. We assume that the H2O molecule lies in the xOz plane and Ox axis is the axis of a dipole moment of a molecule. In this case only ξy13 ¼ ξy31 a0 and ξy23 ¼ ξy32 a0 [13]. The second-order transformation operator S2 was used in the form S2 ¼ S40 þ S31 þ S22 þS13
ðA5Þ
where
1 Snklm ðqk ql qm pn þpn qk ql qm Þ k;l;m;n 2 1 ðq þ Sklm p p p þ p p p q Þ ; k l m n n 2 n k l m
X X α lm 1 α ðq Jα S q q q þ S p p þ p p q Þ ; S31 ¼ klm k l m l m k k k;l;m 2 k i m α
S40 ¼
S22 ¼ S13 ¼
X
X
1X
J J α;β α b 2
X
k;l
αβ l Sk ðqk pl þ pl qk Þ;
Sαβγ J α J β J β qk α;β;γ;k k
The coefficients appearing in Snm may be found in Ref. [13].
56
V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
Appendix B. Some commutation rules For the elementary vibrational operators q and p we use ½pn ; q ¼ ð iÞnpn 1 ; ½p; qn ¼ ð iÞnqn 1
ðB1Þ
For elementary rotational operators ½J z ; J 7 ¼ 7 J 7 ; ½J þ ; J ¼ 2J z
ðB2Þ
In (B2) J 7 ¼ J x 7 iJ y . For elementary rotational operators and rotational matrices ½J z ; Dlμ ¼ μDlμ ; 7
½J 7 ; Dlμ ¼ f l ðμÞDlμ 7 1 ; ½J
2
; Dlμ ¼ lðl þ1ÞDlμ :
ðB3Þ
7 f l ðμÞ ¼ fðl 7 μÞðl 7 μþ 1Þg1=2
and J is the total angular Here momentum of a molecule. These commutators are based on relations (3.120) and (3.123) of Ref. [16]. For l ¼1 Eq. (B3) gives the useful rotational commutator [13] X X X ½iα AJ α ; β Bφβ ¼ φα ε β AJ α γ B α;β β;γ αβγ α
for the direction cosines 1 φx ¼ pffiffiffifD1 1 D11 Þ; 2 i φx ¼ pffiffiffifD1 1 þD11 Þ; 2 φz ¼ D10
ðB4Þ
between the space-fixed direction Z and rotating direction α; εαβγ is the unit anti-symmetrical third-rank tensor, α,β, γ,¼x, y, z and αA and βB are the constants. References [1] Godon M, Bauer A. Helium-broadened widths of the 183 and 380 GHz lines of water vapor. Chem Phys Lett 1988;147:189–91. [2] Goyette TM, De Lucia FC. The pressure broadening of the 313–220 transition of water between 80 and 600 K. J Mol Spectrosc 1990;143: 346–58. [3] Steyert DW, Wang WF, Sirota JM, Donahue NM, Reuter DC. Hydrogen and helium pressure broadening of water transitions in the 380– 600 cm 1 region. J Quant Spectrosc Radiat Transf 2004;83:183–91.
[4] Claveau C, Henry A, Hurtmans D, Valentin A. Narrowing and broadening parameters of H2O lines perturbed by He, Ne, Ar, Kr and nitrogen in the spectral range 1850–2140 cm 1. J Quant Spectrosc Radiat Transf 2001;68:273–98. [5] Claveau C, Valentin A. Narrowing and broadening parameters for H2O lines perturbed by helium, argon and xenon in the 1170– 1440 cm 1 spectral range. Mol Phys 2009;107:1417–22. [6] Solodov AM, Starikov VI. Broadening and shift of lines of the ν2 þν3 band of water vapor induced by helium pressure. Opt Spectrosc 2008;105:14–20. [7] Solodov AM, Starikov VI. Helium-induced halfwidths and line shifts of water vapor transitions of the ν1 þ ν2 and ν2 þν3 bands. Mol Phys 2009;107:43–51. [8] Petrova TM, Solodov AM, Starikov VI, Solodov AA. Measurements and calculations of He-broadening and -shifting parameters of the water vapor transitions of the ν1 þν2 þν3 band. Mol Phys 2012;110: 1493503. [9] Lucchesini A, Gozzini S, Gabbanini C. Water vapor overtones pressure line broadening and shifting measurements. Eur Phys J D 2000;8:223–6. [10] Poddar P, Mitra S, MdM Hossain, Biswas D, Ghosh PN, Ray B. Diode laser spectroscopy of He, N2 and air broadened water vapour transitions belonging to the (2ν1 þν2 þν3) overtone band. Mol Phys 2010;108:1957–64. [11] Petrova TM, Solodov AM, Solodov AA, Starikov VI. Vibrational dependence of an intermolecular potential for H2O–He system. J Quant Spectrosc Radiat Transf 2013241–53. [12] Aliev MR, Watson JKG. Higher-order effects in the vibration-rotation spectra of semirigid molecules. In: Rao KN, editor. Molecular spectroscopy: modern research, v.III. London: Academic Press; 1985. p. 1–67. [13] Camy-Peyret C, Flaud J-M. Vibration–rotation dipole moment operator for asymmetric rotors. In: Rao KN, editor. Molecular spectroscopy: modern research, v.III. London: Academic Press; 1985. p. 69–110. [14] Watson JKG. Determination of centrifugal coefficients of asymmetric top molecules. J Chem Phys 1967;46:1935–49. [15] Watson JKG. Determination of centrifugal coefficients of asymmetric top molecules. II. Dreizler, Dendle, and Rudolph's results. J Chem Phys 1968;48:181–5. [16] Biedenharn LC, Louck JD. Angular momentum in quantum physics. Reading, MA: Addison Wesley; 1981. [17] Labani B, Bonamy J, Robert D, Hartmann J-M, Taine J. Collisional broadening of rotation–vibration lines for asymmetric top molecules. I. Theoretical model for both distant and close collisions. J Chem Phys 1986;84:4256–67. [18] Hoy AR, Mills IM, Strey G. Anharmonic force constant calculations. Mol Phys 1972;24:1265–90. [19] Robert D, Bonamy J. Short range force effects in semiclassical molecular line broadening calculations. J Phys 1979;40:923–43. [20] Bykov AD, Lavrentieva NN, Sinitsa LN. Influence of trajectory model on the line shift in the visible region. Atmos Ocean Opt 1992;5: 907–17. [21] Buldyreva J, Lavrent’eva NN, Starikov VI. Collisional line broadening and shifting of atmosphyric gase. A practical guide for line shape modeling by current semi-classical approaches. London: Imperical College Press; 2010.
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Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Vibration–rotation interaction potential for H2O–A system V.I. Starikov a,b,n a b
Tomsk State University of Control System and Radio Electronics, Tomsk 634050, Russia Tomsk Polytechnic University, Tomsk 634050, Russia
a r t i c l e i n f o
abstract
Article history: Received 11 October 2014 Received in revised form 23 December 2014 Accepted 24 December 2014 Available online 2 January 2015
The contact transformation method is used to derive an effective interaction potential for H2O–A system. The expressions for the second order vibrational and rotational correction terms are obtained. The contributions of the third order, which take into account the Coriolis and Fermi resonances in H2O molecule, are obtained too. The matrix elements of a transformed interaction potential that are necessary for calculating the broadening parameters of H2O lines perturbed by mono-atomic gases are presented. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Perturbation theory Contact transformations Effective interaction potential H2O Interruption functions
1. Introduction Knowledge of the water vapor absorption spectra in the presence of mono-atomic gases is necessary for many applications, mainly for astronomical ones. Water vapor has been detected in many objects of the Solar System, including Venus, Mars, and comets, and was observed in the IR spectra of brown dwarfs, red giant stars, and cold dark molecular clouds. The measured broadening coefficients of H2O molecule in the mixtures with mono-atomic gases Ar, He, Ne, Kr, and Xe are presented in Refs. [1–11]. To calculate the broadening coefficients γ (line-width) and δ (line shift) the interaction potential V(r) (r is the distance between the center of mass of H2O molecule and atom A) as well as rovibrational energies E and wave functions Ψ have to be known. The energies E and functions Ψ must be calculated by solving the Schrodinger equation HΨ ¼EΨ, where H is the Hamiltonian of a vibrating–rotating molecule. In practice the functions Ψ~ instead n Correspondence address: Tomsk State University of Control System and Radio Electronics, Tomsk 634050, Russia. Tel.: þ7 3822 492044; fax: þ7 3822 492086. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.jqsrt.2014.12.018 0022-4073/& 2015 Elsevier Ltd. All rights reserved.
of Ψ are used in calculations of γ and δ. The functions Ψ~ are determined by using the effective Hamiltonian Heff extracted from the transformed Hamiltonian Н~ ¼ U þ HU, where U¼ exp( iS) is the unitary operator of contact transformation and Ψ~ ¼ eiS Ψ . To use the wave functions Ψ~ we have to substitute all matrix elements 〈Ψ|V(r)|Ψ〉 which appear in the calculations of γ and δ for the matrix 0 elements 〈Ψ~ jV~ ðrÞjΨ~ 〉, where V~ ðrÞ ¼ eiS VðrÞe iS is a transformed interaction potential. In Refs. [6,7] the expression for isotropic part, V~ isot , of the transformed interaction potential V~ ðrÞ was obtained. The transformed potential V~ isot depends on the vibrational and rotational operators. This fact explains the unusual rotational dependence of the shift coefficients δ of H2O lines perturbed by He [6–8]. The aim of the present paper is to derive the total expression for the transformed interaction potential V~ ðrÞ for the H2O–A system and the corresponding matrix 0 elements 〈Ψ~ jV~ ðrÞjΨ~ 〉 which are necessary for calculating the broadening coefficients γ and δ. 2. Principle of calculation In the present section only a shortcut presentation is given. The details may be found in Refs. [12,13]. Let V be an
50
V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
interaction potential expanded in order of magnitude with respect to the parameter λ as
3. Initial interaction potential
V ¼ V 0 þ λV 1 þ λ2 V 2 þλ3 V 3 þ…
The interaction potential between H2O molecule and atom A is usually taken in the form of atom–atom potential, which is the sum of pair potentials ! 3 X di ei aa V ¼ V 1;2 ¼ 6 ; ð9Þ r 12 ri i i¼1
ð1Þ
It is necessary to perform on the potential V the same set of contact transformations as those used to go from an initial Hamiltonian H to the transformed Hamiltonian Н~ ¼ eiλS He iλS . When performing the unitary transformation, V is replaced by the transformed potential V~ ¼ eiλS Ve iλS ¼ V þ λ½iS; V þ λ2 =2!½iS; ½iS; V þ …
ð2Þ
The transformation operator S is written in the following form: S ¼ S1 þλS2 þ …
ð3Þ
which, in turn, are modeled by the Lennard–Jones (6)–(12) potentials. In formula (9), the index i refers to the i-th atom of the H2O molecule and di and ei are the atomic pair energy parameters. When expressed in terms of rotational matrices Dlmμ (l r2) this potential formally may be written as Vðr; qÞ ¼
2 l X X l ¼ 0 μ ¼ l
In the series of contact transformations the first, second, etc. transformations are V ð1Þ ¼ eiλS1 Ve iλS1 V ð2Þ ¼ eiλ
2
S2
V ð1Þ e iλ
2
S2
ð4Þ
The transformed potential (2) then is written as ðkÞ ðkÞ V~ ¼ V~ 0 þ V~ 1 þ V~ 2 þ… ¼ V ðkÞ 0 þ V 1 þV 2 þ::
ð5Þ
The upper and lower subscripts represent the number of used transformations and the order of λ, respectively. It is shown in Refs. [6–8] that for an H2O–He system it is necessary to use some terms from a transformed isotropic potential V~ isot , obtained in the third order of perturbation theory. For any commutator [iS, V] in Eq. (2) the relation ½iS; V ¼ ½iS; VV þ λ½iS; V R
ð6Þ
[12,13] is used. Here the subscripts V and R denote the vibrational or rotational commutator, respectively. We assume that the zero-order term V0 does not depend on the vibrational variable and [iS, V0]V ¼ 0. In this case two transformations are required to obtain V~ up to λ3. The contributions into twice-transformed up to λ3 potential (5) take the form
V~ 2 ¼ V 2ð1Þ ¼ V 2 þ ½iS1 ; V 1 V þ ½iS01 ; V 0 R V~ 3 ¼ V 2ð2Þ ¼ V 3 þ ½iS1 ; V ″2 V þ ½iS01 ; V 1 R þ½iS2 ; V 1 V þ ½iS2 ; V 0 R ; ð7Þ where ð8Þ
and S10 ¼S1 þSR. The operator SR was used in Refs. [14,15] to obtain a reduced Hamiltonian for asymmetric top molecules. The formulas for the first and second order transformation operators S0 1 and S2 are presented in Refs. [12,13] and they are presented in Appendix A for convenience. The commutation rules for some vibrational and rotational operators are given in Appendix B.
ð10Þ
Here we assume that the interaction potential (10) depends on the vibrational dimensionless normal coordinates qi(i¼ 1,2,3) of the H2O molecule. The index m¼0 on matrix Dlmμ is omitted. The expression (10) is taken from Ref. [17] where it is written for the functions Vlμ(r,q ¼0). Introducing q – dependence of these functions – we can write ( ) jμj l jμj l 4 X jμj l D12 þ t ðqÞ jμj l E 6 þ t ðqÞ V lμ ðr; qÞ ¼ b c ð11Þ 0 t 0 t r 12 þ t r6 þ t t¼0 The expressions for the quantities ð:::Þ Dð:::Þ ð:::Þ ðq ¼ 0Þ and 0Þ as well as the coefficients b and c are given in Ref. [17]. The quantities D12 þ t, E6 þ t (other indices are omitted) depend on the energy parameters eH eH–A, eO eO–A, dH dH–A, dO dO–A and on the molecular parameters rH1, rH2, rO and 2β which are the H1–G, H2–G, O–G distances and the angle H1GH2, where G is the center of mass of the H2O molecule. These molecular parameters correspond to the instantaneous configuration of a molecule and may be expanded in the normal coordinates qi, for example, as follows: ð:::Þ ð:::Þ Eð:::Þ ðq ¼
r H1 ðqÞ ¼ r H þ
X
r H1;i qi þ
i
V~ 0 ¼ V 0 V~ 1 ¼ V 1
1 V ″2 ¼ V 2 þ ½iS1 ; V 1 V 2
V lμ ðr; qÞDlμ
1X r q q þ… 2 i;j H1;ij i j
ð12Þ
ð:::Þ ð:::Þ In Ref. [17] the quantities ð:::Þ Dð:::Þ Eð:::Þ ðq ¼ 0Þ ð:::Þ ðq ¼ 0Þ and are determined for the equilibrium configuration of the H2O molecule in which the molecular parameters (rH1 ¼ rH2 ¼rH, rO and β) do not depend on qi. The expansions of the molecular parameters rH1, rH2, rO and β in the coordinates qi lead to the expansions of the quantities D12 þ t and E6 þ t in qi
D12 þ t ðqÞ ¼ D12 þ t ð0Þ þ
X
D12 þ t; i qi þ
i
1X D q q þ… 2 i;j 12 þ t; ij i j ð13Þ
and of the functions V lμ ðr; qÞ ¼ V lμ ðrÞ þ
X i
V lμ;i ðrÞqi þ
1X l V ðrÞqi qj þ … 2 i;j μ;ij
ð14Þ
V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
When expansion (14) is substituted, the interaction potential (10) takes the form (1) with X V0 ¼ V l ðrÞDlμ l;μ μ o X nX V1 ¼ V l ðrÞDlμ qi l;μ μ;i i
o 1X nX V2 ¼ V l ðrÞDlμ qi qj l;μ μ;ij 2 i;j
ð15Þ
parameters R V l;α are defined as μ;k x y z
l ¼ 1; μ ¼ 1 pffiffiffi 1 ði= 2ÞSxx k V0 pffiffiffi yy 1 ð1= 2ÞSk V 0 pffiffiffi 1 i 2Sxz 3 V0
l ¼ 2; μ ¼ 2 2 2iSxz 3 V2
2 2iSzz k V2
l ¼ 2; μ ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 iSxx 3=2 V 20 Þ k ðV 2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffi Syy ðV 22 3=2 V 20 Þ k qffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 iSxz 3=2 V 20 Þ 3 ðV 2 þ ð20Þ
In Sections 4–8 the dependence of the potential (10) on the distance r is omitted for brevity.
These parameters satisfy the relations R
4. Once transformed interaction potential (up to λ2) To obtain V~ ¼ V 0 þ V 1 þ V~ 2 it is necessary to calculate two commutators, [iS1,V1]V and [iS0 1,V0]R. The required calculation commutators are given in Appendix B. The substitution S0 1, V0, and V1 in Eq. (7) for V~ 2 results X ij l X l;ij Dl ð V~ μ pi pj þ V~ μ qi qj Þ V~ 2 ¼ l;μ μ i;j X l;α 1X 1X l;αβ V~ μ;k pk þ fDl ; J g fDl ; J J gV~ þ 2 l;μ;α μ α k 2 l;μ;α;β μ α β μ ð16Þ Here {A,B}¼AB þBA is an anticommutator. Other quantities are defined as a sum of two terms ij
51
l V~ μ ¼ ij V lμ þ ij V l;R μ ;
V 1;x 1;k R 2;x V 1;k R 2;z V 1;3
R 1;z ¼ R V 1;x ; R V 1;y ¼ R V 1;y ; R V 1;z 1;3 ¼ V 1;3 ; 1;k 1;k 1;k R 2;x R 2;y R 2;y ¼ R V 2;x ; R V 2;x 2;3 ¼ V 2;3 ; V 1;k ¼ V 1;k ; 1;k R 2;z R 2;z ¼ R V 2;z 1;3 ; V 2;k ¼ V 2;k ;
where k ¼1 or 2. The non-zero parameters R V l;αβ are defined as μ;k pffiffiffi R 1;xz R 1;xz 1 V 1 ¼ V 1 ¼ s111 V 0 = 2; pffiffiffi R 1;yz 1 V 1 ¼ R V 1;xz 1 ¼ is111 V 0 = 2; qffiffiffiffiffiffiffiffiffiffiffiffi ffi R 2;yz 2 V 1 ¼ R V 2;yz 3=2 V 20 Þ; 1 ¼ is111 ðV 2 þ qffiffiffiffiffiffiffiffiffiffiffiffi ffi R 2;xz 2 V 1 ¼ R V 2;xz 3=2 V 20 Þ; 1 ¼ s111 ðV 2 R
V 2;xy ¼ R V 2;xy ¼ 2is111 V 22 2 2
ð21Þ
Formally, all obtained expressions (16)–(21) with l¼ 1 coincide with the corresponding expressions for the P transformed dipole moment operator μ~ Z ¼ φα μ~ α from
l;ij R l; V~ μ ¼ V l;ij μ þ V μ;ij ; l;α þ R V l;α ; V~ μ;k ¼ V l;α μ;k μ;k
α
ð17Þ
Ref. [13] if φα are defined by Eq. (B4) and V10 ¼μz, pffiffiffi V 17 1 ¼ ð 8 μx iμy Þ= 2.
obtained from the vibrational and rotational commutators, respectively. The terms from the vibrational commutator are X ijk l ij l Vμ ¼ 3S V μ;k ;
5. Accidental resonances in H2O molecule and contributions of the third order in the twice-transformed interaction potential
l;αβ R l;αβ V~ μ ¼ V l;αβ μ þ Vμ ;
k
X k l 1 l Sij V μ;k ; V l;ij μ ¼ V μ;ij þ 2 k X α kn l ¼ S V μ;n ; V l;α μ;k n
V l;αβ μ ¼
X
Sαβ V lμ;k k
ð18Þ
k
and The first two terms ij V l;R μ rotational commutator [iS0 1,V0]R are
R
V lμ;ij (17) from the
1y R l S V ; 2 ij μ 1 2Þ R V lμ;ij ¼ y Sij R V lμ ; 2 1Þ ij V l;R μ ¼
where R
V lμ ¼
1 pffiffiffiV 10 δl;1 þ V 22 2
ð19Þ rffiffiffi ! ! 3 2 δμ; 7 1 μ V δ 2 0 l;2
and δl,m is a delta symbol. We take into account that among the coefficients αSij and αSij only yS13 ¼ yS31 a0 and y S23 ¼ yS32 a0 (see Appendix A). The compact expressions for the terms R V l;α and R V l;αβ (17) from the rotational μ;k μ;k commutator [iS0 1,V0]R are not obtained. The non-zero
The rovibrational energies E and wave functions Ψ~ of H2O molecule are usually obtained using the effective Hamiltonian Heff extracted from the transformed Hamiltonian Н~ . The effective Hamiltonian Heff is build for a given polyad (group) of interacting vibrational states. These polyads are formed according to the rule 2v1 þ2v3 þ v2 ¼ p ¼0, 1, 2, 3…., where v1, v3, and v2 are the vibrational quantum numbers. This rule is based on the relation ω1 Eω3 E 2ω2 and it generates the series of accidental resonances in the H2O molecule which have to be taken into account in the transformed isotropic potential [6–8]. In the present study our efforts are concentrated on the Cor Fer contributions V~ 3 and V~ 3 which describe the Coriolis interactions (ω1 Eω3) between vibrational states |2〉¼| v1 þ 1v2v3〉 and |3〉¼|v1v2v3 þ1〉 and Fermi interactions (ω1 E2ω2) between vibrational states |1〉¼|v1v2 þ2v3〉 and |2〉¼|v1 þ1v2v3〉, respectively. These contributions appear in the twice-transformed third-order interaction potential V~ 3 from Eq. (7). The Cor general expression for V~ 3 is X Cor 1X 1 V~ 3 ¼ fDl ; J g i;j α Rlμ;ij ðqi pj þ pj qi Þ 2 l;μ;α μ α 2
ð22Þ
52
V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
matrix elements 〈njV~ jm〉 in which V~ ¼ V 0 þV 1 þ V~ 2 and V0, V1 and V~ 2 are defined by Eqs. (15) and (16), respectively.
where X j l;α 1X α ″ l;jk α jk l α l Rμ;ij ¼ ð Sik V μ Sik V μ Þ 2 Sik V μ;k 2 k k X α kj l R l;α þ2 Si V μ;k þ R V l;α μ;ij þ ð2Þ V μ;ij
ð23Þ
k
and
l,ij l,ij Vμ ¼Vlμ,ij/2 þVμ .
Formulas for R V l;α μ;ij are obtained from
the formulas for R V l;α (20) in which μ;k by
αβ j Si.
Analogously Vlμ,i
R l;α ð2Þ V μ;ij
αβ Sk
must be substituted
are obtained from
Vlμ.
R
V l;α μ;i by
for substituting Taking into account only the isotropic part of interaction potential (l¼0 in Eq. (22)) we can obtain that for H2O molecule Cor V~ 3 ¼ J y ðy R13 q1 p3 þ y R31 q3 p1 Þ;
ð24Þ
0;y R where y Rij ¼ y R00;ij . For this quantity R V 0;y 0;13 ¼ ð2Þ V 0;13 ¼ 0 in Fer (23). The Fermi contribution V~ 3 is extracted from the general expression for X l X Fer 1 ~ l;ij l ~ V~ 3 ¼ V V D q q q þ ðq p p þp p q Þ i j m m i j i j m μ;ijm μ;m μ l;μ i;j;m 2 ð25Þ
6.1. Effective interaction potential in a given vibrational state The effective interaction potential (up to λ2) in the vibrational state |n〉 ¼|v1v2v3〉 is obtained as X X ðnÞ 1 ii ~ l l l ð Vμ D fV þ v þ V~ ¼ 〈njV 0 þ V 1 þ V~ 2 jn〉 ¼ i μ l;μ μ 2 i l;ii 1 þ V~ μ Þg þ 2
ðnÞ V~ ¼
X
l l;12 l;22 Dl ½3V~ μ;122 q1 q22 þ V~ μ;2 ðq2 p1 p2 þp1 p2 q2 Þ þ V~ μ;1 q1 p22 l;μ μ
l 1 V~ μ;ijm ¼ V lμ;ijm þ 2 6 l;ij V~ μ;m ¼
X n
Snmi '' V l;nj μ þ
X n
Sijm V lμ;n þ
1y S V l RV l ; 2 ij μ;m μ
X ijn l;nm 1y l ð6S V μ 4Sjmn in V lμ þ3Sijn S V l RV l m V μ;n Þ þ 4 ij μ;m μ n ð27Þ
6. Vibrational matrix elements of the transformed up to the second-order interaction potential To calculate a line width γif or line shift δif for the rovibrational transition (i)-(f) the matrix elements 〈Ψ~ f jV~ jΨ~ f 〉 and 〈Ψ~ i jV~ jΨ~ i 〉 have to be evaluated. For the functions Ψ~ f and Ψ~ i we can write [13] P P J; K; γ〉 jΨ~ f 〉 ¼ Ψ ðv1 v2 v3 JK a K c Þ〉 ¼ m〉 C ðmÞ k m
l;μ
Dlμ V~ μ ðnÞ þ
l V~ μ ðnÞ ¼ V lμ þ
l
1X l;αβ fDl ; J J gV~ ; 2 l;μ;α;β μ α β μ
X 1 ðvi þ ÞΔV lμ;i ; 2 i
X ϕiik V μ;k 1 V lμ;ii ¼ 2 ωk k l
ΔV lμ;i
ð26Þ In Eq. (26)
l;αβ fDlμ ; J α J β gV~ μ
ð28Þ
k
ð31Þ
ð32Þ
! ð33Þ
and фijk are the cubic anharmonic constants [18] from the intramolecular potential of H2O molecule. The coefficient l l of Dμ in Eq. (31), i.e. V~ μ , may be called the effective vibrational (rotation-free) interaction potential in the state |n〉¼|v1v2v3〉 of H2O molecule. Eq. (32) gives the vibrational correction to the initial potential. It involves the first as well as the second derivatives of the potential with respect to the normal coordinates. The second term in Eq. (31) determines the rotational correction to the effective potenðnÞ tial. It is useful to present V~ (31) in the form ðnÞ ðnÞ ðnÞ V~ ¼ V~ diag þ V~ n:diag
ð34Þ
ðnÞ 〈J,K,γ|V~ n:diag |J,K,γ〉 ¼0.
l
Using approximation 1/2{Dμ, in which l JαJβ}-DμJα Jβ we can obtain a simple equation X ðnÞ ðnÞ l V~ diag ¼ V~ ef f ¼ Dl V~ ðnÞ; ð35Þ l;μ μ μ;ef f where l l l l l l l V~ μ;ef f ðnÞ ¼ V~ μ ðnÞþ V~ μ;rot ¼ V~ μ ðnÞ þ ðJÞ V~ μ U J2 þ ðKÞ V~ μ J 2z þ ðxyÞ V~ μ ðJ 2x J 2y Þ
k
X ð0Þ C k J; K; γ〉 jΨ~ i 〉 ¼ Ψ ð000JK a K c Þ〉 ¼ 0〉
ð30Þ
where
Fer
X
l;μ;α;β
R l Because ii V l;R μ ¼ V μ;ii ¼ 0 in Eq. (17) (see Eq. (19) and Appendix A) then
by the condition 〈1|V~ 3 |2〉 a0. This gives Fer V~ 3 ¼
X
ð36Þ and l 1 1X xx l;yy V~ μ ¼ ðV l;xx ðS þ Syy ÞV lμ;k ; μ þVμ Þ ¼ k 2 2 k k
where the symbol B stands for all the vibrational states of the polyad to which the vibration–rotation state under study belongs and |m〉¼|v1v2v3〉, |0〉 ¼|v1 ¼0 v2 ¼0 v3 ¼0〉. The functions |JKγ〉(γ¼ 71) are defined as pffiffiffi jJKg〉 ¼ fjJK〉 þ γjJ K〉g= 2 jJK ¼ 0; γ〉 ¼ jJK ¼ 0〉; ð29Þ
ðJÞ
where |JK〉 are the symmetric top rotational wave functions. A matrix element 〈Ψ~ f jV~ jΨ~ f 〉 involves the vibrational
ðxyÞ
ðKÞ
l l;zz l V~ μ ¼ V~ μ ðJÞ V~ μ ¼
X
l ðJÞ ~ Szz k V μ;k V μ ; l
ð37Þ
ð38Þ
k
l 1 l;xx l;yy 1X xx l V~ μ ¼ ðV~ μ V~ μ Þ ¼ ðS Sxx k ÞV μ;k 2 2 k k
ð39Þ
V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
The main contribution into the line shift coefficient δif is determined by the first-order interruption function Z
1 1 h ~ ~ ~ S1 ¼ dt 〈Ψ f V isot Ψ f 〉 〈Ψ~ i V~ isot Ψ~ i 〉 ; ð40Þ ℏ 1 which depends on the difference between the isotropic 0 potential V~ isot ¼ V~ 0 in the initial (i) and final (f) rovibrational states. According to Eq. (36) this difference may be presented as 〈Ψ~ f jV~ isot jΨ~ f 〉 〈Ψ~ i jV~ isot jΨ~ i 〉 ¼ ΔV~ vib þ ΔV~ rot ; where ΔV~ vib ¼
ð41Þ
53
involved in Coriolis resonances. In this case Cor Cor Cor 〈Ψ~ f jV~ 3 jΨ~ f 〉 ¼ 〈Ψ~ f 0 jV~ 3 jΨ~ f 0 〉 ¼ R31 f ðv1 ; v3 Þf ðJ; KÞ
where
ð48Þ
R31 ¼1/2{Ry31–Ry13}
and 1=2 f ðv1 ; v3 Þ ¼ ðv1 þ1Þðv3 þ1Þ f
Cor
ðJ; KÞ ¼
X k;k'
0 0 C ð3Þ C ð3Þ C kð2Þ fC ð2Þ 0 g〈JKγ ðiJ y Þ JK γ 〉 k k k0
ð49Þ
In the same way we can obtain the matrix element Fer Fer 0;Ferm 〈Ψ~ f jV~ 3 jΨ~ f 〉 ¼ 〈Ψ~ f 0 jV~ 3 jΨ~ f 0 〉 ¼ 2F 00 f ðv1 ; v2 Þf ðJ; KÞ
ð50Þ
X
v ΔV 00;i ; i ¼ 1;2;3 i
ð42Þ
and X ðmÞ f ðC k Þ2 ðC ð0Þ Þ2 gK 2 k
0X
ΔV~ rot ¼ ðJÞ V~ 0 ½J f ðJ f þ 1Þ J i ðJ i þ 1Þþ ðKÞ V~ 0 0
k
m
ð43Þ (the contribution from the last term of Eq. (36) is omitted). For example, for the ν1 band of H2O molecule in Eq. (42) v1 ¼1, v2 ¼v3 ¼ 0 and ΔV~ vib ¼ ΔV 00;1 . In the approximation of symmetric top molecule Eq. (43) becomes ΔV~ rot ¼ ðJÞ V~ 0 f ðJÞ þ ðKÞ V~ 0 f ðKÞ: 0
0
1=2 f ðv1 ; v2 Þ ¼ ðv2 þ2Þðv2 þ1Þðv3 þ 1Þ X 0;Ferm f ðJ; KÞ ¼ C kð1Þ C ð2Þ k
f ðKÞ ¼ ½K 2f K 2i
ð45Þ
6.2. Off-diagonal vibrational matrix elements Because there are the resonance interactions in the H2O molecule the matrix elements 〈1|V~ |2〉 and 〈2|V~ |3〉 have to be calculated. For the potential V~ 2 (16) the matrix elements 〈1|V~ 2 |2〉 ¼〈v1v2 þ2v3|V~ 2 |v1 þ1v2v32〉¼0 and formally, according to Eq. (16) 〈2jV~ 2 j3〉 ¼ 〈v1 þ1v2 v3 jV~ 2 jv1 v2 v3 þ 1〉 v1 þ1 1=2 v3 þ 1 1=2 X l ¼ Dl V~ l;μ μ μ;1 3 2 2
Fer The term V~ 3 describes the resonance interactions between rovibrational states (f) and (f0 ) with ΔK ¼Kf0 – Kf0 ¼0. Usually they appear for the highly excited rotational states of H2O molecule with K EJ, but the broadening coefficients γ and δ usually are measured for small or middle K. These rotational states may be involved in Fermi resonances with ΔK ¼ 72, 74,… To describe these reso2 2 4 4 nances the rotational operators (J þ þJ ), (J þ þJ ),… have to be introduced into transformed potential. The rotational dependence of Fermi resonance appears in the highly ordered terms of V~ . In the present study the dependence 2 2 of V~ on the operator (J þ þJ ) is introduced in the matrix Fer element 〈1|V~ |2〉 as
3
Fer 〈1jV~ 3 j2〉 ¼ f ðv1 ; v2 ÞfF ð0Þ þF ð2Þ ðJ 2þ þ J 2 Þg
ð46Þ
In this case Fer 0;Fer 2;Fer 〈Ψ~ f jV~ 3 jΨ~ f 〉 ¼ f ðv1 ; v2 Þ½2F ð0Þ f ðJ; KÞ þF ð2Þ f ðJ; KÞ
where X
ϕ13k V lμ;k
k
ðω3 ω1 Þ2 ω2k
ð52Þ
k
ð44Þ
in which K Ka and f ðJÞ ¼ ½J f ðJ f þ 1Þ J i J i þ1 ;
l l V~ μ;1 3 ¼ V~ μ;13 þ
for two states (f) ¼(v1v2 þ2v3)[Jf Kaf Kcf ] and (f0 )¼ (v1 þ1v2v3) [Jf0 Kaf0 Kcf0 ] involved in Fermi resonances. In Eq. (50) only the isotropic part of transformed potential is taken into account,
1 3 0 1 0;22 0;12 F 00 F ð0Þ ¼ pffiffiffi V~ 0;122 V~ 0;1 þ V~ 0;2 ; ð51Þ 2 2 2
þ ðy S13 þ y S13 ÞR V lμ
ð47Þ
where f
2;Fer
ðJ; KÞ ¼
X k
For H2O molecule this term is not equal to zero for l¼1, μ¼ 71. In the initial potential (10) the term with V lμ¼¼17 1 ðr; q ¼ 0Þ is absent and it will be omitted in subsequent calculations. 7. Rovibrational matrix elements of resonance terms Cor Fer The resonance terms V~ 3 and V~ 3 are defined by Eqs. (24) and (26), respectively. Let for the two transitions (i)(f) and (i0 )-(f0 ) the upper rovibrational states (f)¼ (v1 þ1v2v3)[JfKafKcf] and (f0 )¼(v1v2v3 þ1)[Jf0 Kaf0 Kcf0 ] be
ð53Þ
ðC ð1Þ C kð2Þ7 2 þ C kð2Þ C kð1Þ7 2 Þf½JðJ þ 1Þ k
KðK 7 1ÞJðJ þ 1Þ ðK 7 1ÞðK 72Þg1=2
ð54Þ
8. Interruption functions To calculate the line broadening coefficients, γ and δ, the interruption functions of the first, S1(rc,v), and second, S2(rc,v), order have to be known [17,19,21] (rc is the closest approaching distance between colliding partners and v is their relative velocity). The relations between the function ð:::Þ ð:::Þ S2(rc,v) and the quantities ð:::Þ Dð:::Þ Eð:::Þ ðq ¼ 0Þ of ð:::Þ ðq ¼ 0Þ and
54
V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
the initial interaction potential V(r,q¼0) (10) and (11) are established in Refs. [17,21]. With the little modifications they may be adopted to the effective potential (35). According to Eqs. (11), (14) and (15) the functions Vlμ(r), Vlμ,i(r), and Vlμ,ij(r) may be presented as ! þt þt 4 X V l;6 V l;12 μ μ 6 þ t þ 12 þ t ; V lμ ðrÞ ¼ r r t¼0 ! l;6 þ t þt 4 X V μ;i V l;12 μ;i V lμ;i ðrÞ ¼ 6 þ t þ 12 þ t ; r r t¼0 ! l;6 þ t þt 4 X V μ;ij V l;12 μ;ij l 6 þ t þ 12 þ t ð55Þ V μ;ij ðrÞ ¼ r r t¼0 When these functions are substituted, the effective potenðnÞ tial V~ ef f (35) becomes 0 1 ðnÞ ~ l;6 þ t ðnÞ ~ l;12 þ t 4 X X Vμ Vμ ðnÞ l @ V~ ef f ðrÞ ¼ ð56Þ D þ 12 þ t A; l;μ μ r6 þ t r t¼0 where, according to (36), ðnÞ
in expression (33) for ΔV~ vib , in expressions (37)–(39) Cor for ΔV~ rot , in formulas (17) and (23) for ΔCp , and in Fer formula (51) for ΔCp all derivatives Vlμ(r),Vlμ,i(r), and Vlμ, l,p l,p l,p ij(r) must be substituted by the derivatives Vμ ,Vμ,i, and Vμ,ij, respectively. Formula (59) may be complicated in an application. Usually the expansion (11) (with l¼ 0) for the isotropic potential is modeled by the function V mod isot ¼
C 6 C 12 þ ; r 6 r 12
in which C6 ¼4εσ6, and C12 ¼ 4εσ12, and ε and σ are known parameters. In Refs. [6–8] a simplified form of S1 which mod corresponds to Visot (62), was used in the calculation of H2O line shift in the H2O–He system. This form depends on some variable parameters. The simplification and parameterization of S1 (59) may be performed by placing in (59) Cor Fer ΔC8 ¼ΔC10 ¼ΔC14 ¼ΔC16 ¼ΔC12 ¼ ΔC12 ¼0 so that " ! # 3π ΔC rot ΔC Cor ΔC Fer 21 ΔC vib vib 6 6 6 12 þ a12 S1 ¼ ΔC 6 a6 1 þ þ þ 6 8vℏr 5c 32 ΔC vib ΔC vib ΔC vib ΔC vib 6 6 6 6 rc
ð63Þ
l;p l;p l;p l;p l;p l;p l;p V~ μ ¼ V~ μ ðnÞ þ V~ μ;rot ðnÞ ¼ V~ μ ðnÞ þ ðJÞ V~ μ J2 þ ðKÞ V~ μ J 2z þ ðxyÞ V~ μ ðJ 2x J 2y Þ
ð57Þ l;p V~ μ ðnÞ,
l;p …ðxyÞ V~ μ
and p¼6 þt or 12 þt. The expressions for are given by (32), (33), (37)–(39) in which the quantities Vlμ(r),Vlμ,i(r), and Vlμ,ij(r) must be substituted by the quantities Vl,pμ, Vl,pμ,i, and Vl,pμ,ij, respectively. The isotropic potential (l ¼0 in Eq. (56)) may be rewritten in the form ! ðnÞ ðnÞ X C~ 6 þ t C~ 12 þ t ðnÞ þ ð58Þ V~ isot ðrÞ ¼ t ¼ 0;2;4 r 6 þ t r 12 þ t ðnÞ 0;p with C~ p ¼ ðnÞ V~ 0 . The interruption function of the first order S1(rc,v) is defined by Eq. (40). The substitution of Eq. (58) in Eq. (40) results in the following: 3π 5 ΔC 8 35 ΔC 10 21 ΔC 12 a6 ΔC 6 a8 a10 þa12 S1 ¼ 5 6 r 2c 48 r 4c 32 r 6c 8vℏr c 77 ΔC 14 143 ΔC 16 þ a14 þ a16 ð59Þ 128 r 8c 256 r 10 c
The coefficients ap are defined in the way that a6 ¼(8/ 3π)A6,… a12 ¼(8/3π) (32/21) A12, … a16 ¼(8/3π) (256/143) A16 and Z 1 dy Ap ¼ ð60Þ pffiffiffi yp=2 fy 1 þ V isot ðr c Þ yV isot ð yr c Þg1=2 1 In the calculation of integral (40) the kinematical model of Ref. [19] (taking into account the close collisions) and the exact trajectories [20] for the colliding particles are used. In the straight line approximation for the trajectory Visot(rc)¼0 in Eq. (60) and all coefficients ap ¼1 in (59). Each contribution ΔCp from Eq. (59) is written as rot Cor Fer ΔC p ¼ ΔC vib p þ ΔC p þΔC p þΔC p vib ΔCp
ð61Þ
rot ΔCp
and are defined by expresHere the quantities sions (42) and (43) for ΔV~ vib and ΔV~ rot , respectively, Cor Fer Cor Fer ΔCp ¼ 〈Ψ~ f jV~ jΨ~ f 〉 (48) and ΔCp ¼〈Ψ~ f jV~ jΨ~ f 〉 (53). But 3
3
ð62Þ
The often-used expression C6 ¼[μ2 þ3/2uα]α2 relates C6 to the molecular parameter. Here u¼u1u2/(u1 þu2), μ and α are the dipole moment and the averaged polarizability of H2O molecule, α2 is the polarizability of an atom A and u1 and u2 are the ionization energies of the molecule and vib atom, respectively. This expression allows us to write ΔC6 as 2 2 ΔC vib 6 ¼ ½μðf Þ μðiÞ þ3=2uðαðf Þ αðiÞÞα2 ;
ð64Þ
where μ(f), μ(i) and α(f), α(i) are the dipole moments and the averaged polarizability of H2O molecule in the final (v1v2v3) and initial (000) vibrational states, respectively. 0;6 The variable parameters may be introduced as xðJÞ ¼ ðJÞ V~ = ΔC vib 6 ,
6 6 vib vib ðKÞ vib ðKÞ ~ 0;6 6 x6 ¼ V 6 =ΔC 6 and y¼ΔC12 /(σ ΔC6 ). In the Cor vib 6 ðJ; KÞ=ΔC vib the function ΔC Cor 6 =ΔC 6 ¼ R31 f ðv1; v3 Þf 6 Cor
ratio fCor(J,K) may be approximated by the function f (J,K)¼ Cor Cor Cor C1 F(Jf,Kf) C% , in which C1 is some constant, F(Jf,Kf)¼ Cor 1/2 {Jf(Jf þ1) Kf (Kf þ1)} and C% determine the mixing of the final states involved in Coriolis resonances. With a Cor
vib 6 variable parameter r 631 ¼ C Cor the ratio ΔC6 / 1 R31 =ΔC 6 vib ΔC6 becomes vib Cor 6 ΔC Cor 6 =ΔC 6 ¼ r 31 f ðv1 ; v3 ÞFðJ f ; K f ÞC %
ð65Þ
In the same way in (52)–(54) we can approximate f0, Fe 0,Fer Fer ,2,Ferm Ferm (J,K)¼C0 C% , f2,Fer(J,K)¼C2 C% F2(Jf,Kf), where C0 Ferm 0,Fer 2,Fer and C2 are some constants and C% and C% determine the mixing of the final states, involved in Fermi resonances with ΔKf ¼0, 72, respectively, F2(Jf,Kf) ¼F(Jf, Kf) F(Jf, Kf þ1). The introduction of the variable parameters by the expressions
Fer
0;6Þ vib ð0Þ ¼ 2C Fer r ð12 0 F =ΔC 6 ; 2;6Þ vib ð2Þ ¼ C Fer r ð12 2 F =ΔC 6
V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
55
allows us to rewrite Eq. (63) in the form
Appendix A. Transformation operators
n 3π ðKÞ Cor Cor S1 ¼ ΔC vib a6 ð1 þ xðJÞ 6 6 f ðJÞ þ x6 f ðKÞ þ r 31 f ðv1 ; v3 ÞFðJ f ; K f ÞC % 8vℏr 5c
S01
h i 21 σ 6 ð2;6Þ 0;Fer 2;Fer f ðv C þr F ðJ ; K ÞC ; v Þ þ a y þ ðrð0;6Þ 2 1 2 12 f f % % 12 12 32 r c
ð66Þ To modify the relations [17,21] between the interruption functions S2(rc,v) and the potential parameters it is useful to rewrite an effective potential (56) in the form 0 1 jμj ~ l jμj ~ l 4 X X ðnÞ jμj l D 12 þ t A jμj l E 6 þ t l ~ @ ; ð67Þ V ef f ðrÞ ¼ D 0 ct 6 þ t þ 0 bt 12 þ t l;μ μ r r t¼0 in which jμj ~ l D 12 þ t
l;12 þ t 2 l;12 þ t 2 J2 þ ðKÞ V~ μ J z þ ðxyÞ V~ μ ðJ x J 2y Þ
o
ð68Þ
This formula determines the vibrational and rotational ~ ð:::Þ and ð:::Þ E~ ð:::Þ . In the dependence of the quantities ð:::Þ D ð:::Þ ð:::Þ same way this formula determines the vibrational and rotational dependence of the interruption functions S2 involved in the calculations of the broadening coefficients γ and δ. In formulas for the functions S2(rc,v) [17,21] the ~ ð:::Þ and ð:::Þ E~ ð:::Þ have to be used instead of quantities ð:::Þ D ð:::Þ
Dð:::Þ ð:::Þ
and
ð:::Þ ð:::Þ ð:::Þ Eð:::Þ .
¼ S1 þSR ¼ S30 þ S21 þS12 þ SR
ðA1Þ
The first and second indices in S1 show the degree of the vibrational (qk or pk ¼ i∂/∂qk) and rotational (Jα) operators, respectively; qk are the dimensionless normal coordinates of H2O molecule (k ¼1, 2, 3) and Jα(α¼ x, y, z) represent the angular momentum components of the H2O molecule in the molecular axis system. Here [12,13] X kij 1 S pk pi pj þ Sjki ðqk qi pj þpj qk qi Þ S30 ¼ k;i;j 2 X αβ S12 ¼ S J J p α;β;k k α β k S21 ¼
( X 1 jμj l jμj l þt 1 ¼ 0 D12 þ t þ ð0 b12 þ t Þ vi þ ΔV l;12 μ;i 2 i l;12 þ t þ ðJÞ V~ μ
The first-order transformation operator S10 is written as
1X J ðα S q q þ α Skl pk pl Þ 2 k;l;α α kl k l
SR ¼ s111 ðJ x J y J z þJ z J y J x Þ ¼
ðA2Þ
i is111 h 2 J þ ðJ z þ 1Þ ðJ z þ 1ÞJ 2 2
and Sαβ ¼ ð Bαβ =ωk Þ; k k Skij ¼ фkij ωk ωi ωj =ð3Ωkij Þ; Sjki ¼ фkij ωj ðω2j ω2k ω2i Þ=ð2Ωkij Þ; α α
Skl ¼ 4Bα ξαkl ðωk ωl Þ1=2 =ðω2k ω2l Þ; Skl ¼
α
Skk ¼ 0;
2Bα ξαkl ðω2k þ ω2l Þ=½ðωk ωl Þ1=2 ðω2l ω2l Þ;
The analysis of measured broadening coefficients of H2O lines [6–8,11], perturbed by He, revealed some new details. Firstly it was shown that the shift coefficients δ depend on the sign of ΔKa ¼Kaf–Kai in the transition (i)(f), (i) (000) [JiKaiKci], (f) (v1v2v3) [JfKafKcf]. Secondly, the calculated line shifts coefficients δ for two transitions (i)-(f) and (i0 )-(f0 ) depend on the existence of the resonance interactions between upper rovibrational states (f) and (f0 ). These details were explained by the rotational dependence of the first-order interruption function S1(rc,v). It is shown in the present paper that the second-order interruption functions S2(rc,v) used in the calculations of the broadening coefficients γ and δ are vibrational and rotational dependent too. This dependence is connected with the rovibrational dependence of an effective interaction potential (which is generated by the vibrational dependence of the intra-molecular parameters rH, rO, and β of H2O molecule) and with the use of the effective wave functions. The application of the obtained expressions for the effective interaction potential of the concrete H2O–A system will be discussed in the next paper.
α
Skk ¼ 0;
Ωkij ¼ ðωk þ ωj þωi Þð ωk þ ωj þωi Þðωk ωj þ ωi Þðωk þ ωj ωi Þ ðA4Þ
ð:::Þ
9. Conclusion
ðA3Þ
αβ
In these equations Bα and Bk are the rotational constants and rotational derivatives, respectively, ωk are the frequencies of the harmonic vibrations. Modifications of the formulas (A4) in the case of Coriolis and Fermi resonances are discussed in [13]. We assume that the H2O molecule lies in the xOz plane and Ox axis is the axis of a dipole moment of a molecule. In this case only ξy13 ¼ ξy31 a0 and ξy23 ¼ ξy32 a0 [13]. The second-order transformation operator S2 was used in the form S2 ¼ S40 þ S31 þ S22 þS13
ðA5Þ
where
1 Snklm ðqk ql qm pn þpn qk ql qm Þ k;l;m;n 2 1 ðq þ Sklm p p p þ p p p q Þ ; k l m n n 2 n k l m
X X α lm 1 α ðq Jα S q q q þ S p p þ p p q Þ ; S31 ¼ klm k l m l m k k k;l;m 2 k i m α
S40 ¼
S22 ¼ S13 ¼
X
X
1X
J J α;β α b 2
X
k;l
αβ l Sk ðqk pl þ pl qk Þ;
Sαβγ J α J β J β qk α;β;γ;k k
The coefficients appearing in Snm may be found in Ref. [13].
56
V.I. Starikov / Journal of Quantitative Spectroscopy & Radiative Transfer 155 (2015) 49–56
Appendix B. Some commutation rules For the elementary vibrational operators q and p we use ½pn ; q ¼ ð iÞnpn 1 ; ½p; qn ¼ ð iÞnqn 1
ðB1Þ
For elementary rotational operators ½J z ; J 7 ¼ 7 J 7 ; ½J þ ; J ¼ 2J z
ðB2Þ
In (B2) J 7 ¼ J x 7 iJ y . For elementary rotational operators and rotational matrices ½J z ; Dlμ ¼ μDlμ ; 7
½J 7 ; Dlμ ¼ f l ðμÞDlμ 7 1 ; ½J
2
; Dlμ ¼ lðl þ1ÞDlμ :
ðB3Þ
7 f l ðμÞ ¼ fðl 7 μÞðl 7 μþ 1Þg1=2
and J is the total angular Here momentum of a molecule. These commutators are based on relations (3.120) and (3.123) of Ref. [16]. For l ¼1 Eq. (B3) gives the useful rotational commutator [13] X X X ½iα AJ α ; β Bφβ ¼ φα ε β AJ α γ B α;β β;γ αβγ α
for the direction cosines 1 φx ¼ pffiffiffifD1 1 D11 Þ; 2 i φx ¼ pffiffiffifD1 1 þD11 Þ; 2 φz ¼ D10
ðB4Þ
between the space-fixed direction Z and rotating direction α; εαβγ is the unit anti-symmetrical third-rank tensor, α,β, γ,¼x, y, z and αA and βB are the constants. References [1] Godon M, Bauer A. Helium-broadened widths of the 183 and 380 GHz lines of water vapor. Chem Phys Lett 1988;147:189–91. [2] Goyette TM, De Lucia FC. The pressure broadening of the 313–220 transition of water between 80 and 600 K. J Mol Spectrosc 1990;143: 346–58. [3] Steyert DW, Wang WF, Sirota JM, Donahue NM, Reuter DC. Hydrogen and helium pressure broadening of water transitions in the 380– 600 cm 1 region. J Quant Spectrosc Radiat Transf 2004;83:183–91.
[4] Claveau C, Henry A, Hurtmans D, Valentin A. Narrowing and broadening parameters of H2O lines perturbed by He, Ne, Ar, Kr and nitrogen in the spectral range 1850–2140 cm 1. J Quant Spectrosc Radiat Transf 2001;68:273–98. [5] Claveau C, Valentin A. Narrowing and broadening parameters for H2O lines perturbed by helium, argon and xenon in the 1170– 1440 cm 1 spectral range. Mol Phys 2009;107:1417–22. [6] Solodov AM, Starikov VI. Broadening and shift of lines of the ν2 þν3 band of water vapor induced by helium pressure. Opt Spectrosc 2008;105:14–20. [7] Solodov AM, Starikov VI. Helium-induced halfwidths and line shifts of water vapor transitions of the ν1 þ ν2 and ν2 þν3 bands. Mol Phys 2009;107:43–51. [8] Petrova TM, Solodov AM, Starikov VI, Solodov AA. Measurements and calculations of He-broadening and -shifting parameters of the water vapor transitions of the ν1 þν2 þν3 band. Mol Phys 2012;110: 1493503. [9] Lucchesini A, Gozzini S, Gabbanini C. Water vapor overtones pressure line broadening and shifting measurements. Eur Phys J D 2000;8:223–6. [10] Poddar P, Mitra S, MdM Hossain, Biswas D, Ghosh PN, Ray B. Diode laser spectroscopy of He, N2 and air broadened water vapour transitions belonging to the (2ν1 þν2 þν3) overtone band. Mol Phys 2010;108:1957–64. [11] Petrova TM, Solodov AM, Solodov AA, Starikov VI. Vibrational dependence of an intermolecular potential for H2O–He system. J Quant Spectrosc Radiat Transf 2013241–53. [12] Aliev MR, Watson JKG. Higher-order effects in the vibration-rotation spectra of semirigid molecules. In: Rao KN, editor. Molecular spectroscopy: modern research, v.III. London: Academic Press; 1985. p. 1–67. [13] Camy-Peyret C, Flaud J-M. Vibration–rotation dipole moment operator for asymmetric rotors. In: Rao KN, editor. Molecular spectroscopy: modern research, v.III. London: Academic Press; 1985. p. 69–110. [14] Watson JKG. Determination of centrifugal coefficients of asymmetric top molecules. J Chem Phys 1967;46:1935–49. [15] Watson JKG. Determination of centrifugal coefficients of asymmetric top molecules. II. Dreizler, Dendle, and Rudolph's results. J Chem Phys 1968;48:181–5. [16] Biedenharn LC, Louck JD. Angular momentum in quantum physics. Reading, MA: Addison Wesley; 1981. [17] Labani B, Bonamy J, Robert D, Hartmann J-M, Taine J. Collisional broadening of rotation–vibration lines for asymmetric top molecules. I. Theoretical model for both distant and close collisions. J Chem Phys 1986;84:4256–67. [18] Hoy AR, Mills IM, Strey G. Anharmonic force constant calculations. Mol Phys 1972;24:1265–90. [19] Robert D, Bonamy J. Short range force effects in semiclassical molecular line broadening calculations. J Phys 1979;40:923–43. [20] Bykov AD, Lavrentieva NN, Sinitsa LN. Influence of trajectory model on the line shift in the visible region. Atmos Ocean Opt 1992;5: 907–17. [21] Buldyreva J, Lavrent’eva NN, Starikov VI. Collisional line broadening and shifting of atmosphyric gase. A practical guide for line shape modeling by current semi-classical approaches. London: Imperical College Press; 2010.