Charge asymmetry in the rovibrationally excited HD molecule Nikita Kirnosov, Keeper Sharkey, and Ludwik Adamowicz Citation: The Journal of Chemical Physics 140, 104115 (2014); doi: 10.1063/1.4867912 View online: http://dx.doi.org/10.1063/1.4867912 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Charge asymmetry in rovibrationally excited HD+ determined using explicitly correlated all-particle Gaussian functions J. Chem. Phys. 139, 204105 (2013); 10.1063/1.4834596 Properties of the B+-H2 and B+-D2 complexes: A theoretical and spectroscopic study J. Chem. Phys. 137, 124312 (2012); 10.1063/1.4754131 The Na + – H 2 cation complex: Rotationally resolved infrared spectrum, potential energy surface, and rovibrational calculations J. Chem. Phys. 129, 184306 (2008); 10.1063/1.3005785 Quantum control of molecular vibrational and rotational excitations in a homonuclear diatomic molecule: A full three-dimensional treatment with polarization forces J. Chem. Phys. 124, 014111 (2006); 10.1063/1.2141616 Relaxation behavior of rovibrationally excited H 2 in a rarefied expansion J. Chem. Phys. 121, 9876 (2004); 10.1063/1.1807819
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THE JOURNAL OF CHEMICAL PHYSICS 140, 104115 (2014)
Charge asymmetry in the rovibrationally excited HD molecule Nikita Kirnosov,1 Keeper Sharkey,2 and Ludwik Adamowicz1,2 1 2
Department of Physics, University of Arizona, Tucson, Arizona 85721, USA Department of Chemistry and Biochemistry, University of Arizona, Tucson, Arizona 85721, USA
(Received 17 December 2013; accepted 25 February 2014; published online 13 March 2014) The recently developed method for performing all-particle non-Born-Oppenheimer variational calculations on diatomic molecular systems excited to the first excited rotational state and simultaneously vibrationally excited is employed to study the charge asymmetry and the level lifetimes of the HD molecule. The method uses all-particle explicitly correlated Gaussian functions. The nonlinear parameters of the Gaussians are optimized with the aid of the analytical energy gradient determined with respect to these parameters. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4867912] I. INTRODUCTION
As it can be shown even with the simplest theoretical model, the average nucleus–electron distance differs in the hydrogen and deuterium atoms due to the larger mass of the nucleus in the latter. Even though this effect is subtle (on the order of ×10−4 bohr for H and D atoms1 ), it also appears in the HD molecule, where the electron has a slightly larger reduced mass while being close to deuteron than while being close to the proton. This results in charge asymmetry in the HD molecule. The effect is purely nonadiabatic and can only be studied within an approach where the Born-Oppenheimer (BO) approximation is not assumed. The asymmetric electron behavior in HD results in appearance of a small dipole moment. We have studied the HD dipole moment in the ground state in one of our previous works2 and the dipole-moment value obtained there agreed very well with the experimental value of 3.19(±0.055) × 10−4 a.u.3 Due to the non-zero dipole moment, pure rotational transitions in the HD molecule are visible in the experiment, though they are very weak. The charge asymmetry makes the HD molecule an interesting model for testing high-level quantum-mechanical methods. The calculations performed with such methods can be of interest to the astrophysics community, as HD is one of the most important molecular systems present in the interstellar space. In our previous study, the HD charge asymmetry in the rotational ground state as a function of the vibrational quantum number was considered.1 The non-BO calculations performed in that work showed that the asymmetry reaches maximum for the ninth (ν = 8) vibrational excited state and then decreases for higher states to eventually approach zero at the dissociation. In the present study, we investigate the effect of rotation excitation on the HD charge asymmetry. In the non-BO approach employed in present work, we use all-particle explicitly correlated Gaussian functions (ECGs).4, 5 In these functions, the interparticle correlation effects are described through the dependence of the Gaussian exponents on the inter-particle distances. In addition, the Gaussians are multiplied by powers of the internuclear distance (for diatomics) or distances (for molecules with more
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than two nuclei) to describe the inter-nuclear correlation. This correlation in much stronger than the inter-electron correlation because, as the nuclei are much heavier than the electrons, they avoid each other to much greater extent than the light electrons. This effect needs to be properly described in the wave function and the powers of the internuclear distance(s) facilitate this description. The Hamiltonian (called the internal Hamiltonian) used in our non-BO calculations is obtained by rigorously separating out the operator representing the kinetic energy of the center-of-mass motion from the laboratory-frame nonrelativistic Hamiltonian. The internal Hamiltonian represents the motion of pseudoparticles around the center of the internal coordinate system where the charge of the reference particle is located. We call the particles described by the internal Hamiltonian pseudoparticles because, while they have the same charges as the original particles, their masses are reduced masses (see Sec. II) not the original masses. One can call the system represented by the internal Hamiltonian a “generalized atom,” because it describes motion of charged particles (for a molecule some of these charges are equal to −1, but some charges corresponding to pseudonuclei are positive) in the central potential created by the charge of the reference particle. As the internal Hamiltonian commutes with the square of the total angular momentum (Nˆ 2 ) and its z component (Nˆ z ), the manifold of the internal-Hamiltonian eigenfunctions can be divided into subsets, each corresponding to a different Nˆ 2 quantum number, N. The eigenfunctions with N = 0 represent rotationless vibrational states. The eigenfunctions with N = 1 represent states where the rotational motion of the nuclei is excited to the first excited state. This again is only an approximation because angular excitations of the electrons can also contribute, although very little (as our previous calculations in Refs. 6 and 7 have shown), to the N = 1 wave functions. In view of the above, it is clear that direct variational single-step non-BO calculations of molecular bound ground and, particularly, excited state is not a simple matter. In the first part of this work, we describe the method we used in the calculations (a more complete description of the method can be found in our recent reviews8, 9 ). The discussion
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© 2014 AIP Publishing LLC
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of the results obtained in the calculations is presented in the second part. II. THE METHOD
As mentioned, the total non-relativistic laboratory-frame Hamiltonian of the system can be rigorously separated into an operator representing the motion of the center of mass and an operator representing the system’s internal state. We perform this separation by making a transformation from the laboratory Cartesian coordinates, Ri , i = 1, . . . , Ntot , where Ntot is the number of particles (nuclei+electrons) in the system, to an internal reference frame with origin at a selected reference particle (particle one): ri = Ri+1 − R1 . This transformation to the internal coordinates together with the conjugate momentum transformation yields the following nonadiabatic Hamiltonian representing the internal state of the system (in atomic units): ⎞ ⎛ n n 1 1 1 ∇2 + ∇ ∇r ⎠ Hˆ = − ⎝ 2 i=1 μi ri i=1,j =1,i=j m0 ri j +
n q0 qi i=1
+
ri
n qi qj , r j >i=1 ij
(1)
where m0 is the mass of particle one (the reference particle; in the present calculations the deuteron), mi , i = 1, . . . , Ntot − 1 are masses of particles 2, . . . , Ntot , qi , i = 0, . . . , Ntot − 1 are charges of particles 1, . . . , Ntot , ∇ri is the gradient with respect to the x, y, and z coordinates of ri , and n = Ntot − 1 (n = 3 in the HD calculations). In the calculations presented in this work, m0 is the mass of the deuteron and m1 is the mass of the proton (md = 3670.4829652me and mp = 1836.15267261me ,10 where me is the electron mass), and m2 and m3 are the electron masses. The reduced masses, μi , are defined as: μi = m0 mi /(m0 + mi ). The potential energy is the same as in the laboratory-frame Hamiltonian, but is now written using the internal coordinates, rij = ||rj − ri || = ||Rj + 1 − Ri + 1 || and rj = ||rj || = ||Rj + 1 − R1 ||. The symbol prime ( ) is used for vector/matrix transposition. More information on the nonadiabatic internal Hamiltonian and on the center-of-mass transformation can be found in Refs. 11 and 12. The spatial part of the HD non-BO wave functions of the N = 1 vibrational states are expanded in terms of one-center, spherically symmetric ECGs multiplied by even powers (pk ) of the internuclear distance, r1 , and by the x1 coordinate of the r1 vector13–16 φk = x1 r1 k exp[−r (Ak ⊗ I3 )r], p
{r1 , r2 , r3 } .
(2)
The pk powers in our calculations where r = p range from 0 to 250. The r1 k factors in functions (2) allow for describing the nucleus–nucleus correlations and for generating radial nodes in the wave function when the molecule becomes vibrationally excited. Ak in (2) is a symmetric matrix of exponential parameters. To make the Gaussians (2) squareintegrable, Ak is represented in a Cholesky factorized form as Ak = Lk Lk , where Lk is a lower triangular matrix. With this representation, Ak is automatically positive definite for any
real values of the Lk matrix elements. ⊗ in (2) is the Kronecker product symbol. Before functions (2) are used in expanding the wave functions of HD they are symmetrized with respect to the permutation of the (pseudo)electron labels, as the considered states are electronic singlet states. Matrix elements of Lk and the pk powers are the variational parameters which are optimized by the energy minimization. In this minimization, the analytical energy gradient determined with respect to the Lk matrix elements is employed. This greatly expedites the calculations and allows achieving high accuracy of the results. The capability for performing non-BO variational calculations for diatomic systems with basis functions (2) was recently developed.7 At present in basis functions (2) we only include the x1 angular factor. However, in general, basis functions with factors xi , where i = 2, . . . , n should also appear as they may provide some contributions to the wave functions of the N = 1 states. However, as the ECGs with the xi , i = 2, . . . , n, describe contributions where, instead of rotational excitation of the nuclei (which are described by ECGs with the x1 factor) the i electron becomes excited to a higher angular-momentum state, these contributions are expected to be very small. This is because the electronic excitations are usually much higher in terms of energy than the rotational excitations. Also, it should be mentioned that ECGs (2), even though they do not explicitly include the xi , i = 2, . . . , n, angular factors, they effectively include some contributions of the xi , i = 2, . . . , n, ECGs because the x1 ECGs and the xi ECGs (i = 2, . . . , n) are not orthogonal if their Ak matrices have non-zero off-diagonal elements. However, it should be noted that this is certainly not the most effective way of including these contributions. A better way would be to include them explicitly in the basis set. The main focus of the present calculations is to investigate how the HD charge asymmetry is affected by a rotational excitation. For such purpose, the basis functions (2) should suffice. They should also provide very good energies for the considered states. However, to converge the state energies to virtually exact values, the xi , i = 2, . . . , n, ECGs are probably needed. Work is currently is progress in our laboratory to include such functions in the calculations of N = 1 states of diatomic molecules. The aim of this work is to calculate the whole vibrational spectrum of the HD molecule in the first excited rotational state (N = 1). The calculations are performed with the variational method applied separately to each state. In the variational optimization, the total internal energy of the state expressed as the expectation value of the Hamiltonian (1) is minimized. The minimization is performed with respect to the linear expansion coefficients, {ck }, by solving the generalized eigenvalue problem, and with respect to the Gaussian parameters, {pk , Lk }, E = min
c H ({pk } , {Lk })c . c S({pk } , {Lk })c
(3)
As mentioned the analytical energy gradient calculated with respect to the Lk matrix elements is used in the minimization. There are a number of minimization strategies one can consider using. The one, which we find effective and which allows for better control (and elimination) of linear
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dependencies between ECGs that may arise in the minimization, involves optimization of Lk of one basis function at a time. In this optimization, each time the Lk parameters of the optimized function are changed, the generalized eigenvalue problem is resolved to assure that the total energy is an upper bound to the exact nonrelativistic energy of the state considered in the calculation. There are 18 bound vibrational states of HD corresponding to N = 1. The ECG basis for each state is generated by taking the basis set of 10 000 ECGs for the corresponding N = 0 states and reoptimizing the Lk matrix elements of the Gaussians. No reoptimization of the pk powers is performed. In the next step, 1000-3000 additional ECGs are added to the basis set of each state. More functions are used for higher states than for lower states. The addition of new functions is done stepwise by adding functions one-by-one in groups of 100. After each function is added to the basis set its Lk parameters and the pk power is optimized. After the addition of each 100 functions the whole basis set is reoptimized by cycling over all ECGs and reoptimizing their Lk parameters with the gradient-based approach. At each optimization step the ECGs are checked for linear dependencies, and, if they appear, they are removed. Next, after the generation of the basis sets for all considered N = 1 states is completed, the non-BO nonrelativistic wave functions are used to calculate the expectation values for the following interparticle distances and their squares: the deuteron–proton distance, rdp = r1 , the deuteron–electron distance, rde = r2 , the proton–electron distance, rpe = r12 , the electron–electron distance, and ree = r23 . A comparison of the rde and rpe expectation values allows for determining the degree of the charge asymmetry in the different states of HD. The non-BO wave functions are also used to calculate the deuteron-proton correlation function (i.e., the one-particle relative density of the proton with respect to the deuteron associated with the coordinate r1 ). The correlation function is defined as17 ∞ gi (ξ ) = i (r)|δ(r1 − ξ )|i (r) =
|i (ξ , r2 , r3 )|2 dr2 dr3 ,
−∞
(4) where δ(r1 − ξ ) is the three-dimensional Dirac delta function. As in the non-BO calculations both electrons and nuclei are treated on equal footing, the only information on the molecular structure is obtained in the form of expectation values of the structural parameters. The nucleus-nucleus correlation functions also provide some information on the structure of the system in different states. Another important quantity which is evaluated are the transition dipole moments corresponding to the N ← N + 1 deexcitations (ψ i denotes the wave function of the initial N + 1 state and ψ f is the wave function of the final N state, M is a total mass of the system) ⎞ ⎛ n n m j ⎠ ⎝ ψi |xj |ψf . qp δpj − ψi |Tˆ |ψf = M j =1 p=0
(5)
For the HD molecule, this general expression reduces to ψi |TˆH D |ψf =
n
(−1)δ1j ψi |xj |ψf ,
(6)
j =1
where ψ i |xj |ψ f can be easily evaluated.18 While being an experimentally measurable quantity, the transition moment by itself does not provide much physical insight. Thus, in the next step we use it to calculate the dipole oscillator strength – a dimensionless quantity related to the probability of the transition – defined as fi→f =
n 8π rj Y1m (ˆrj )|ψf |2 . (Ei − Ef )|ψi | 3 j =1
(7)
Performing the manipulations described in Ref. 19 and integrating over the angular coordinates, we obtain fi→f =
2 3(2Nf + 1)
(Ei − Ef )|ψi |Tˆ |ψf |2 .
(8)
In order to calculate the states lifetimes, we need to consider the transition probability per unit time Wi→f = 2α 3
2Nf + 1 (Ei − Ef )2 fi→f , 2Nf + 3
(9)
where α is a fine structure constant and factor (2Nf + 1)/(2Nf + 3) is introduced to reverse the transitions into the more intuitive N + 1 → N notation. The level lifetime is then calculated as ⎛ ⎞−1 Wi→f ⎠ . (10) τi = ⎝ f (Ef 0 and ξ y > 0, and the ξ z coordinate set to zero.
approach where the BO approximation is not assumed, the vibrational quantum number, ν, is, strictly speaking, not. A particular non-BO state considered here, apart from the dominant adiabatic contribution, which is a product of the electronic ground-state wave function and the particular nuclear vibrational and rotational wave functions, also may contain some small contributions from products which involve electronic excited states and different vibrational states. The protondeuteron density (also called the proton-deuteron correlation function) may show the coupling of these states, for example, it did for the highest excited N = 0 and N = 1 states of HD+ .21 However, the effect is usually very small and hard to notice on the density plot.
The N = 1 proton-deuteron density function for HD are shown for some selected lowest, intermediate, and highest vibrational states in Figure 1. Let us examine plot (a) in the figure. It shows the deuteron-proton density for the lowest N = 1 vibrational state plotted for the xy plane with z coordinate set to zero. As the wave function is antisymmetric with respect to the reflection in the x = 0 plane, the density shown in the plot is zero along the y axis and has two maxima located along the x axis, one at positive x’s and one at negative x. These maxima show where the probability of finding the proton relative to the deuteron is the highest. As the density is symmetric with respect to x → −x and y → −y reflections, it is sufficient to only show the density in the positive
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TABLE III. Some expectation values corresponding to the interparticle distances calculated for the N = 1, ν = 0, 1, . . . , 17 rovibrational states of the HD molecule. rdp is the deuteron–proton distance, rpe is the proton–electron distance, rde is the deuteron–electron distance, and ree is the electron–electron distance. All values are in a.u. ν
rdp
rpe
rde
ree
r2dp
r2pe
r2de
r2ee
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1.443860 1.527156 1.613422 1.703172 1.797055 1.895914 2.000849 2.113345 2.235428 2.369941 2.521025 2.694934 2.901631 3.158197 3.497136 3.992217 4.880354 11.33121
1.572337 1.616230 1.661495 1.708372 1.757173 1.808300 1.862282 1.919837 1.981945 2.049988 2.125977 2.212954 2.315761 2.442699 2.609559 2.852196 3.286050 6.467008
1.572047 1.615935 1.661195 1.708069 1.756866 1.807990 1.861969 1.919522 1.981629 2.049673 2.125665 2.212645 2.315460 2.442408 2.609282 2.851937 3.285815 6.466802
2.198018 2.253515 2.311979 2.374001 2.440327 2.511914 2.589997 2.676219 2.772791 2.882765 3.010507 3.162488 3.348832 3.586565 3.907542 4.383023 5.240088 11.53383
2.109055 2.406117 2.729372 3.082389 3.469731 3.897351 4.373131 4.907835 5.516468 6.220626 7.052629 8.063051 9.335955 11.02294 13.43158 17.31976 25.44194 152.8215
3.134499 3.323204 3.524212 3.739364 3.971032 4.222324 4.497371 4.801830 5.143589 5.533985 5.989945 6.537844 7.221277 8.118345 9.386286 11.40949 15.57409 79.40179
3.133389 3.322056 3.523025 3.738139 3.969769 4.221025 4.496038 4.800467 5.142201 5.532580 5.988531 6.536434 7.219887 8.116996 9.385005 11.40831 15.57305 79.40095
5.787216 6.089125 6.414403 6.767526 7.154231 7.581995 8.060678 8.603611 9.229152 9.963297 10.84420 11.93026 13.31623 15.16859 17.81630 22.04935 30.67284 158.7655
(ξ x > 0, ξ y > 0) quadrant of the xy plane. This is shown in plot (b) for the lowest N = 1 vibrational state. Using the same “quadrant” convention, the densities for states ν = 7, 13, and 17 are presented in plots (c)-(e) in the figure. These plots show the increasing number of density oscillations with the vibrational excitation. The presence of the r1m factors in the ECG basis functions is key to describe these oscillations. In order to investigate charge asymmetry appearing in the HD molecule, we need to evaluate the expectation values of the interparticle distances for all vibrational states with single rotational excitation. These values, as well as expectation values of squares of interparticle distances are presented in Table III. Figure 2 shows the difference between the average proton–electron and deuteron–electron distances as a function of vibrational excitation. According to the plot, the ν = 8 state has the highest charge asymmetry and the ν = 17 has the lowest asymmetry. As the results suggest, even though the HD
permanent dipole moment is very small, in relative terms, it may change by as much as 50% as the molecule deexcites from ν = 17 to ν = 8. Figure 3 shows the oscillator strengths computed for the (N = 0, ν = i) → (N = 1, ν = i) and (N = 0, ν = i) → (N = 1, ν = i − 1) transitions. Columns f( ν = 0) and f( ν = +1) of Table II contain numerical values of the plotted data. While the oscillator strength monotonically decreases for the ν = 0 transitions, the ν = +1 oscillator strength reaches a maximum for the ν = 11 state and then decreases for the higher states. The oscillator strengths for the ν = +1 transitions become greater than the oscillator strengths for the ν = 0 transitions at ν = 8, where the charge asymmetry is at maximum. The lifetimes of the HD (N = 0, ν = i), i = 1, . . . , 17, rotationless states calculated using the oscillator strengths of the (N = 0, ν = i) → (N = 1, ν = i − 1), (N = 0, ν = i) → (N = 1, ν = i − 2), . . . , and (N = 0, ν = i) → (N = 1,
FIG. 2. The difference between the average proton–electron and deuteron– electron distances, rpe − rde .
FIG. 3. Oscillator strengths for the pure rotational N = 1 → N = 0 transitions.
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TABLE IV. Dipole transition moments of R(0) and P(1) branches of ν(0 → ν ) transitions. All values are in 10−4 D. R(0) Ref. \ ν 24 25 26 27 28 29 Present
P(1)
0
1
2
3
1
2
0.19(2)
0.0795(35)
0.450(30) 0.435(11)
0.17(2)
8.36 8.306 8.463 8.560 8.885
0.515(20) 0.504(12) 0.598 0.560 0.552 0.5579 0.5472
0.160 0.192 0.200 0.2022 0.1703
0.100 0.082 0.0870 0.0878 0.0696
0.485 0.466 0.4708 0.4567
0.176 0.179 0.1805 0.1481
ν = 0) transitions are shown in Figure 4. It is assumed that the lifetimes are primarily dependent on the N = 1 transitions. The values used in the plot can be found in Table II. Interestingly, the longest-living state is the (N = 0, ν = 17) state. The highest longevity of this state is due to the oscillator strength for the (N = 0, ν = 17) → (N = 1, ν = 16) transition being considerably smaller than that for the corresponding transitions for the ν < 17 states. Due to the lack of experimental data concerning the state lifetimes for HD, as well as due to the lack of high-level calculations of these quantities, there is no direct way to verify our results. Such verification was possible in our previous study of HD+ molecular ion,18 where the high-quality results of Tian22 and Pilon23 were available. Nevertheless, a less direct approach can be still used to evaluate the quality of our predictions. Since the oscillator strength decreases almost by an order of magnitude as ν increases by one (the ν = 0 and ν = 1 cases are an exception), the lifetime of a particular state is almost fully defined by the transition-dipole moments of not more than the three lowest transitions in Eq. (5). Table IV shows a comparison of the transition dipole moments of the ν(0 → ν ) transitions calculated in this work and the available experimental (Refs. 24 and 25) and theoretical BO (Refs. 26–29) results. As it can be seen, the present study yields results which are in a very good agreement with the experiment and with the most recent theoretical calculations. This indicates that the curve which shows the lifetime results presented in Figure 4 is accurate and reliable.
3
0.076 0.0742 0.0749 0.0613
IV. SUMMARY
In this work, a non-BO approach is used to study rovibrational states of the HD molecule corresponding to single rotational excitation (N = 1 states). The complete N = 1 rovibrational spectrum is calculated. The dissociation energies and the transition energies are also determined for N = 0 and N = 1 states and compared with recent results obtained with a method based on the BO approximation. The comparison shows a very good agreement of the two sets of results and indicates that the ECG basis set used in the present calculations can reliably describe states with low and high oscillatory behavior. This oscillatory behavior of the HD rovibrational states considered in this work is demonstrated using plots of the proton-deuteron correlation functions. In the present calculations, we also determined expectation values of the interparticle distances. These values are used to analyze the charge asymmetry in HD. The analysis shows that, as expected, the rotational excitation, elongates the average internuclear distance, but it has no significant effect on the charge asymmetry pattern of the N = 1 states as compared to the N = 0 states. Since according to the selection rule the N = 0 rovibrational states can only deexcite to the N = 1 states with smaller vibrational quantum numbers, the present calculation provides sufficient data for evaluating the interstate transition moments and lifetimes of the N = 0 states. The obtained lifetimes plotted as a function of vibrational quantum number show a similar behavior as those obtained for the HD+ ion in Ref. 18. ACKNOWLEDGMENTS
We are grateful to the University of Arizona Research Computing Center for the use of their computer resources. This work has been supported in part by the National Science Foundation (NSF) through the graduate research fellowship awarded to Keeper L. Sharkey; Grant No. DGE1-1143953. 1 S. Bubin, F. Leonarski, M. Stanke, and L. Adamowicz, J. Chem. Phys. 130,
124120 (2009). Cafiero and L. Adamowicz, Phys. Rev. Lett. 89, 073001 (2002). 3 Z. Lu, G. C. Tabisz, and L. Ulivi, Phys. Rev. A 47, 1159 (1993). 4 S. Bubin, M. Pavanello, W.-Ch. Tung, K. L. Skarkey, and L. Adamowicz, Chem. Rev. 113, 36 (2013). 5 J. Mitroy, S. Bubin, W. Horiuchi, Y. Suzuki, L. Adamowicz, W. Cencek, K. Szalewicz, J. Komasa, D. Blume, and K. Varga, Rev. Mod. Phys. 85, 693 (2013). 2 M.
FIG. 4. Lifetimes of the rotationless states.
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L. Sharkey, N. Kirnosov, and L. Adamowicz, J. Chem. Phys. 139, 164119 (2013). 7 K. L. Sharkey, N. Kirnosov, and L. Adamowicz, Phys. Rev. A 88, 032513 (2013). 8 M. Cafiero, S. Bubin, and L. Adamowicz, Phys. Chem. Chem. Phys. 5, 1491 (2003). 9 S. Bubin, M. Cafiero, and L. Adamowicz, Adv. Chem. Phys. 131, 377 (2005). 10 P. J. Mohr, B. N. Taylor, and D. B. Newell, J. Phys. Chem. Ref. Data 37(3), 1187 (2008). 11 D. B. Kinghorn and R. D. Poshusta, Phys. Rev. A 47, 3671 (1993). 12 D. B. Kinghorn and R. D. Poshusta, Int. J. Quantum Chem. 62, 223 (1997). 13 D. B. Kinghorn and L. Adamowicz, J. Chem. Phys. 110, 7166 (1999). 14 D. B. Kinghorn and L. Adamowicz, Phys. Rev. Lett. 83, 2541 (1999). 15 S. Bubin and L. Adamowicz, J. Chem. Phys. 118, 3079 (2003). 16 M. Pavanello, S. Bubin, M. Molski, and L. Adamowicz, J. Chem. Phys. 123, 104306 (2005). 17 S. Bubin and L. Adamowicz, Chem. Phys. Lett. 403, 185 (2005).
J. Chem. Phys. 140, 104115 (2014) 18 N.
Kirnosov, K. L. Sharkey, and L. Adamowicz, Phys. Rev. A 89, 012513 (2014). 19 Z. C. Yan, J. F. Babb, and A. Dalgarno, Phys. Rev. A 54, 2824 (1996). 20 K. Pachucki and J. Komasa, Phys. Chem. Chem. Phys. 12, 9188 (2010). 21 N. Kirnosov, K. L. Sharkey, and L. Adamowicz, J. Chem. Phys. 139, 204105 (2013). 22 Q. L. Tian, L. Y. Tang, Z. X. Zhong, Z. C. Yan, and T. Y. Shi, J. Chem. Phys. 137, 024311 (2012). 23 H. O. Pilon and D. Baye, Phys. Rev. A 88, 032502 (2013). 24 A. R. W. McKellar, Can. J. Phys. 51, 389 (1973); Astrophys. J. 185, L53 (1973); Can. J. Phys. 52, 1144 (1974). 25 N. H. Rich, J. W. C. Johns, and A. R. W. McKellar, J. Mol. Spectrosc. 95, 432 (1982). 26 L. Wolniewicz, Can. J. Phys. 53, 1207 (1975); 54, 672 (1976). 27 A. L. Ford and J. C. Browne, Phys. Rev. A 16, 1992 (1977). 28 W. R. Thorson, J. H. Choi, and S. K. Knudson, Phys. Rev. A 131, 22 (1985); 131, 34 (1985). 29 K. Pachucki and J. Komasa, Phys. Rev. A 78, 052503 (2008).
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THE JOURNAL OF CHEMICAL PHYSICS 140, 104115 (2014)
Charge asymmetry in the rovibrationally excited HD molecule Nikita Kirnosov,1 Keeper Sharkey,2 and Ludwik Adamowicz1,2 1 2
Department of Physics, University of Arizona, Tucson, Arizona 85721, USA Department of Chemistry and Biochemistry, University of Arizona, Tucson, Arizona 85721, USA
(Received 17 December 2013; accepted 25 February 2014; published online 13 March 2014) The recently developed method for performing all-particle non-Born-Oppenheimer variational calculations on diatomic molecular systems excited to the first excited rotational state and simultaneously vibrationally excited is employed to study the charge asymmetry and the level lifetimes of the HD molecule. The method uses all-particle explicitly correlated Gaussian functions. The nonlinear parameters of the Gaussians are optimized with the aid of the analytical energy gradient determined with respect to these parameters. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4867912] I. INTRODUCTION
As it can be shown even with the simplest theoretical model, the average nucleus–electron distance differs in the hydrogen and deuterium atoms due to the larger mass of the nucleus in the latter. Even though this effect is subtle (on the order of ×10−4 bohr for H and D atoms1 ), it also appears in the HD molecule, where the electron has a slightly larger reduced mass while being close to deuteron than while being close to the proton. This results in charge asymmetry in the HD molecule. The effect is purely nonadiabatic and can only be studied within an approach where the Born-Oppenheimer (BO) approximation is not assumed. The asymmetric electron behavior in HD results in appearance of a small dipole moment. We have studied the HD dipole moment in the ground state in one of our previous works2 and the dipole-moment value obtained there agreed very well with the experimental value of 3.19(±0.055) × 10−4 a.u.3 Due to the non-zero dipole moment, pure rotational transitions in the HD molecule are visible in the experiment, though they are very weak. The charge asymmetry makes the HD molecule an interesting model for testing high-level quantum-mechanical methods. The calculations performed with such methods can be of interest to the astrophysics community, as HD is one of the most important molecular systems present in the interstellar space. In our previous study, the HD charge asymmetry in the rotational ground state as a function of the vibrational quantum number was considered.1 The non-BO calculations performed in that work showed that the asymmetry reaches maximum for the ninth (ν = 8) vibrational excited state and then decreases for higher states to eventually approach zero at the dissociation. In the present study, we investigate the effect of rotation excitation on the HD charge asymmetry. In the non-BO approach employed in present work, we use all-particle explicitly correlated Gaussian functions (ECGs).4, 5 In these functions, the interparticle correlation effects are described through the dependence of the Gaussian exponents on the inter-particle distances. In addition, the Gaussians are multiplied by powers of the internuclear distance (for diatomics) or distances (for molecules with more
0021-9606/2014/140(10)/104115/8/$30.00
than two nuclei) to describe the inter-nuclear correlation. This correlation in much stronger than the inter-electron correlation because, as the nuclei are much heavier than the electrons, they avoid each other to much greater extent than the light electrons. This effect needs to be properly described in the wave function and the powers of the internuclear distance(s) facilitate this description. The Hamiltonian (called the internal Hamiltonian) used in our non-BO calculations is obtained by rigorously separating out the operator representing the kinetic energy of the center-of-mass motion from the laboratory-frame nonrelativistic Hamiltonian. The internal Hamiltonian represents the motion of pseudoparticles around the center of the internal coordinate system where the charge of the reference particle is located. We call the particles described by the internal Hamiltonian pseudoparticles because, while they have the same charges as the original particles, their masses are reduced masses (see Sec. II) not the original masses. One can call the system represented by the internal Hamiltonian a “generalized atom,” because it describes motion of charged particles (for a molecule some of these charges are equal to −1, but some charges corresponding to pseudonuclei are positive) in the central potential created by the charge of the reference particle. As the internal Hamiltonian commutes with the square of the total angular momentum (Nˆ 2 ) and its z component (Nˆ z ), the manifold of the internal-Hamiltonian eigenfunctions can be divided into subsets, each corresponding to a different Nˆ 2 quantum number, N. The eigenfunctions with N = 0 represent rotationless vibrational states. The eigenfunctions with N = 1 represent states where the rotational motion of the nuclei is excited to the first excited state. This again is only an approximation because angular excitations of the electrons can also contribute, although very little (as our previous calculations in Refs. 6 and 7 have shown), to the N = 1 wave functions. In view of the above, it is clear that direct variational single-step non-BO calculations of molecular bound ground and, particularly, excited state is not a simple matter. In the first part of this work, we describe the method we used in the calculations (a more complete description of the method can be found in our recent reviews8, 9 ). The discussion
140, 104115-1
© 2014 AIP Publishing LLC
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of the results obtained in the calculations is presented in the second part. II. THE METHOD
As mentioned, the total non-relativistic laboratory-frame Hamiltonian of the system can be rigorously separated into an operator representing the motion of the center of mass and an operator representing the system’s internal state. We perform this separation by making a transformation from the laboratory Cartesian coordinates, Ri , i = 1, . . . , Ntot , where Ntot is the number of particles (nuclei+electrons) in the system, to an internal reference frame with origin at a selected reference particle (particle one): ri = Ri+1 − R1 . This transformation to the internal coordinates together with the conjugate momentum transformation yields the following nonadiabatic Hamiltonian representing the internal state of the system (in atomic units): ⎞ ⎛ n n 1 1 1 ∇2 + ∇ ∇r ⎠ Hˆ = − ⎝ 2 i=1 μi ri i=1,j =1,i=j m0 ri j +
n q0 qi i=1
+
ri
n qi qj , r j >i=1 ij
(1)
where m0 is the mass of particle one (the reference particle; in the present calculations the deuteron), mi , i = 1, . . . , Ntot − 1 are masses of particles 2, . . . , Ntot , qi , i = 0, . . . , Ntot − 1 are charges of particles 1, . . . , Ntot , ∇ri is the gradient with respect to the x, y, and z coordinates of ri , and n = Ntot − 1 (n = 3 in the HD calculations). In the calculations presented in this work, m0 is the mass of the deuteron and m1 is the mass of the proton (md = 3670.4829652me and mp = 1836.15267261me ,10 where me is the electron mass), and m2 and m3 are the electron masses. The reduced masses, μi , are defined as: μi = m0 mi /(m0 + mi ). The potential energy is the same as in the laboratory-frame Hamiltonian, but is now written using the internal coordinates, rij = ||rj − ri || = ||Rj + 1 − Ri + 1 || and rj = ||rj || = ||Rj + 1 − R1 ||. The symbol prime ( ) is used for vector/matrix transposition. More information on the nonadiabatic internal Hamiltonian and on the center-of-mass transformation can be found in Refs. 11 and 12. The spatial part of the HD non-BO wave functions of the N = 1 vibrational states are expanded in terms of one-center, spherically symmetric ECGs multiplied by even powers (pk ) of the internuclear distance, r1 , and by the x1 coordinate of the r1 vector13–16 φk = x1 r1 k exp[−r (Ak ⊗ I3 )r], p
{r1 , r2 , r3 } .
(2)
The pk powers in our calculations where r = p range from 0 to 250. The r1 k factors in functions (2) allow for describing the nucleus–nucleus correlations and for generating radial nodes in the wave function when the molecule becomes vibrationally excited. Ak in (2) is a symmetric matrix of exponential parameters. To make the Gaussians (2) squareintegrable, Ak is represented in a Cholesky factorized form as Ak = Lk Lk , where Lk is a lower triangular matrix. With this representation, Ak is automatically positive definite for any
real values of the Lk matrix elements. ⊗ in (2) is the Kronecker product symbol. Before functions (2) are used in expanding the wave functions of HD they are symmetrized with respect to the permutation of the (pseudo)electron labels, as the considered states are electronic singlet states. Matrix elements of Lk and the pk powers are the variational parameters which are optimized by the energy minimization. In this minimization, the analytical energy gradient determined with respect to the Lk matrix elements is employed. This greatly expedites the calculations and allows achieving high accuracy of the results. The capability for performing non-BO variational calculations for diatomic systems with basis functions (2) was recently developed.7 At present in basis functions (2) we only include the x1 angular factor. However, in general, basis functions with factors xi , where i = 2, . . . , n should also appear as they may provide some contributions to the wave functions of the N = 1 states. However, as the ECGs with the xi , i = 2, . . . , n, describe contributions where, instead of rotational excitation of the nuclei (which are described by ECGs with the x1 factor) the i electron becomes excited to a higher angular-momentum state, these contributions are expected to be very small. This is because the electronic excitations are usually much higher in terms of energy than the rotational excitations. Also, it should be mentioned that ECGs (2), even though they do not explicitly include the xi , i = 2, . . . , n, angular factors, they effectively include some contributions of the xi , i = 2, . . . , n, ECGs because the x1 ECGs and the xi ECGs (i = 2, . . . , n) are not orthogonal if their Ak matrices have non-zero off-diagonal elements. However, it should be noted that this is certainly not the most effective way of including these contributions. A better way would be to include them explicitly in the basis set. The main focus of the present calculations is to investigate how the HD charge asymmetry is affected by a rotational excitation. For such purpose, the basis functions (2) should suffice. They should also provide very good energies for the considered states. However, to converge the state energies to virtually exact values, the xi , i = 2, . . . , n, ECGs are probably needed. Work is currently is progress in our laboratory to include such functions in the calculations of N = 1 states of diatomic molecules. The aim of this work is to calculate the whole vibrational spectrum of the HD molecule in the first excited rotational state (N = 1). The calculations are performed with the variational method applied separately to each state. In the variational optimization, the total internal energy of the state expressed as the expectation value of the Hamiltonian (1) is minimized. The minimization is performed with respect to the linear expansion coefficients, {ck }, by solving the generalized eigenvalue problem, and with respect to the Gaussian parameters, {pk , Lk }, E = min
c H ({pk } , {Lk })c . c S({pk } , {Lk })c
(3)
As mentioned the analytical energy gradient calculated with respect to the Lk matrix elements is used in the minimization. There are a number of minimization strategies one can consider using. The one, which we find effective and which allows for better control (and elimination) of linear
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dependencies between ECGs that may arise in the minimization, involves optimization of Lk of one basis function at a time. In this optimization, each time the Lk parameters of the optimized function are changed, the generalized eigenvalue problem is resolved to assure that the total energy is an upper bound to the exact nonrelativistic energy of the state considered in the calculation. There are 18 bound vibrational states of HD corresponding to N = 1. The ECG basis for each state is generated by taking the basis set of 10 000 ECGs for the corresponding N = 0 states and reoptimizing the Lk matrix elements of the Gaussians. No reoptimization of the pk powers is performed. In the next step, 1000-3000 additional ECGs are added to the basis set of each state. More functions are used for higher states than for lower states. The addition of new functions is done stepwise by adding functions one-by-one in groups of 100. After each function is added to the basis set its Lk parameters and the pk power is optimized. After the addition of each 100 functions the whole basis set is reoptimized by cycling over all ECGs and reoptimizing their Lk parameters with the gradient-based approach. At each optimization step the ECGs are checked for linear dependencies, and, if they appear, they are removed. Next, after the generation of the basis sets for all considered N = 1 states is completed, the non-BO nonrelativistic wave functions are used to calculate the expectation values for the following interparticle distances and their squares: the deuteron–proton distance, rdp = r1 , the deuteron–electron distance, rde = r2 , the proton–electron distance, rpe = r12 , the electron–electron distance, and ree = r23 . A comparison of the rde and rpe expectation values allows for determining the degree of the charge asymmetry in the different states of HD. The non-BO wave functions are also used to calculate the deuteron-proton correlation function (i.e., the one-particle relative density of the proton with respect to the deuteron associated with the coordinate r1 ). The correlation function is defined as17 ∞ gi (ξ ) = i (r)|δ(r1 − ξ )|i (r) =
|i (ξ , r2 , r3 )|2 dr2 dr3 ,
−∞
(4) where δ(r1 − ξ ) is the three-dimensional Dirac delta function. As in the non-BO calculations both electrons and nuclei are treated on equal footing, the only information on the molecular structure is obtained in the form of expectation values of the structural parameters. The nucleus-nucleus correlation functions also provide some information on the structure of the system in different states. Another important quantity which is evaluated are the transition dipole moments corresponding to the N ← N + 1 deexcitations (ψ i denotes the wave function of the initial N + 1 state and ψ f is the wave function of the final N state, M is a total mass of the system) ⎞ ⎛ n n m j ⎠ ⎝ ψi |xj |ψf . qp δpj − ψi |Tˆ |ψf = M j =1 p=0
(5)
For the HD molecule, this general expression reduces to ψi |TˆH D |ψf =
n
(−1)δ1j ψi |xj |ψf ,
(6)
j =1
where ψ i |xj |ψ f can be easily evaluated.18 While being an experimentally measurable quantity, the transition moment by itself does not provide much physical insight. Thus, in the next step we use it to calculate the dipole oscillator strength – a dimensionless quantity related to the probability of the transition – defined as fi→f =
n 8π rj Y1m (ˆrj )|ψf |2 . (Ei − Ef )|ψi | 3 j =1
(7)
Performing the manipulations described in Ref. 19 and integrating over the angular coordinates, we obtain fi→f =
2 3(2Nf + 1)
(Ei − Ef )|ψi |Tˆ |ψf |2 .
(8)
In order to calculate the states lifetimes, we need to consider the transition probability per unit time Wi→f = 2α 3
2Nf + 1 (Ei − Ef )2 fi→f , 2Nf + 3
(9)
where α is a fine structure constant and factor (2Nf + 1)/(2Nf + 3) is introduced to reverse the transitions into the more intuitive N + 1 → N notation. The level lifetime is then calculated as ⎛ ⎞−1 Wi→f ⎠ . (10) τi = ⎝ f (Ef 0 and ξ y > 0, and the ξ z coordinate set to zero.
approach where the BO approximation is not assumed, the vibrational quantum number, ν, is, strictly speaking, not. A particular non-BO state considered here, apart from the dominant adiabatic contribution, which is a product of the electronic ground-state wave function and the particular nuclear vibrational and rotational wave functions, also may contain some small contributions from products which involve electronic excited states and different vibrational states. The protondeuteron density (also called the proton-deuteron correlation function) may show the coupling of these states, for example, it did for the highest excited N = 0 and N = 1 states of HD+ .21 However, the effect is usually very small and hard to notice on the density plot.
The N = 1 proton-deuteron density function for HD are shown for some selected lowest, intermediate, and highest vibrational states in Figure 1. Let us examine plot (a) in the figure. It shows the deuteron-proton density for the lowest N = 1 vibrational state plotted for the xy plane with z coordinate set to zero. As the wave function is antisymmetric with respect to the reflection in the x = 0 plane, the density shown in the plot is zero along the y axis and has two maxima located along the x axis, one at positive x’s and one at negative x. These maxima show where the probability of finding the proton relative to the deuteron is the highest. As the density is symmetric with respect to x → −x and y → −y reflections, it is sufficient to only show the density in the positive
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TABLE III. Some expectation values corresponding to the interparticle distances calculated for the N = 1, ν = 0, 1, . . . , 17 rovibrational states of the HD molecule. rdp is the deuteron–proton distance, rpe is the proton–electron distance, rde is the deuteron–electron distance, and ree is the electron–electron distance. All values are in a.u. ν
rdp
rpe
rde
ree
r2dp
r2pe
r2de
r2ee
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1.443860 1.527156 1.613422 1.703172 1.797055 1.895914 2.000849 2.113345 2.235428 2.369941 2.521025 2.694934 2.901631 3.158197 3.497136 3.992217 4.880354 11.33121
1.572337 1.616230 1.661495 1.708372 1.757173 1.808300 1.862282 1.919837 1.981945 2.049988 2.125977 2.212954 2.315761 2.442699 2.609559 2.852196 3.286050 6.467008
1.572047 1.615935 1.661195 1.708069 1.756866 1.807990 1.861969 1.919522 1.981629 2.049673 2.125665 2.212645 2.315460 2.442408 2.609282 2.851937 3.285815 6.466802
2.198018 2.253515 2.311979 2.374001 2.440327 2.511914 2.589997 2.676219 2.772791 2.882765 3.010507 3.162488 3.348832 3.586565 3.907542 4.383023 5.240088 11.53383
2.109055 2.406117 2.729372 3.082389 3.469731 3.897351 4.373131 4.907835 5.516468 6.220626 7.052629 8.063051 9.335955 11.02294 13.43158 17.31976 25.44194 152.8215
3.134499 3.323204 3.524212 3.739364 3.971032 4.222324 4.497371 4.801830 5.143589 5.533985 5.989945 6.537844 7.221277 8.118345 9.386286 11.40949 15.57409 79.40179
3.133389 3.322056 3.523025 3.738139 3.969769 4.221025 4.496038 4.800467 5.142201 5.532580 5.988531 6.536434 7.219887 8.116996 9.385005 11.40831 15.57305 79.40095
5.787216 6.089125 6.414403 6.767526 7.154231 7.581995 8.060678 8.603611 9.229152 9.963297 10.84420 11.93026 13.31623 15.16859 17.81630 22.04935 30.67284 158.7655
(ξ x > 0, ξ y > 0) quadrant of the xy plane. This is shown in plot (b) for the lowest N = 1 vibrational state. Using the same “quadrant” convention, the densities for states ν = 7, 13, and 17 are presented in plots (c)-(e) in the figure. These plots show the increasing number of density oscillations with the vibrational excitation. The presence of the r1m factors in the ECG basis functions is key to describe these oscillations. In order to investigate charge asymmetry appearing in the HD molecule, we need to evaluate the expectation values of the interparticle distances for all vibrational states with single rotational excitation. These values, as well as expectation values of squares of interparticle distances are presented in Table III. Figure 2 shows the difference between the average proton–electron and deuteron–electron distances as a function of vibrational excitation. According to the plot, the ν = 8 state has the highest charge asymmetry and the ν = 17 has the lowest asymmetry. As the results suggest, even though the HD
permanent dipole moment is very small, in relative terms, it may change by as much as 50% as the molecule deexcites from ν = 17 to ν = 8. Figure 3 shows the oscillator strengths computed for the (N = 0, ν = i) → (N = 1, ν = i) and (N = 0, ν = i) → (N = 1, ν = i − 1) transitions. Columns f( ν = 0) and f( ν = +1) of Table II contain numerical values of the plotted data. While the oscillator strength monotonically decreases for the ν = 0 transitions, the ν = +1 oscillator strength reaches a maximum for the ν = 11 state and then decreases for the higher states. The oscillator strengths for the ν = +1 transitions become greater than the oscillator strengths for the ν = 0 transitions at ν = 8, where the charge asymmetry is at maximum. The lifetimes of the HD (N = 0, ν = i), i = 1, . . . , 17, rotationless states calculated using the oscillator strengths of the (N = 0, ν = i) → (N = 1, ν = i − 1), (N = 0, ν = i) → (N = 1, ν = i − 2), . . . , and (N = 0, ν = i) → (N = 1,
FIG. 2. The difference between the average proton–electron and deuteron– electron distances, rpe − rde .
FIG. 3. Oscillator strengths for the pure rotational N = 1 → N = 0 transitions.
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TABLE IV. Dipole transition moments of R(0) and P(1) branches of ν(0 → ν ) transitions. All values are in 10−4 D. R(0) Ref. \ ν 24 25 26 27 28 29 Present
P(1)
0
1
2
3
1
2
0.19(2)
0.0795(35)
0.450(30) 0.435(11)
0.17(2)
8.36 8.306 8.463 8.560 8.885
0.515(20) 0.504(12) 0.598 0.560 0.552 0.5579 0.5472
0.160 0.192 0.200 0.2022 0.1703
0.100 0.082 0.0870 0.0878 0.0696
0.485 0.466 0.4708 0.4567
0.176 0.179 0.1805 0.1481
ν = 0) transitions are shown in Figure 4. It is assumed that the lifetimes are primarily dependent on the N = 1 transitions. The values used in the plot can be found in Table II. Interestingly, the longest-living state is the (N = 0, ν = 17) state. The highest longevity of this state is due to the oscillator strength for the (N = 0, ν = 17) → (N = 1, ν = 16) transition being considerably smaller than that for the corresponding transitions for the ν < 17 states. Due to the lack of experimental data concerning the state lifetimes for HD, as well as due to the lack of high-level calculations of these quantities, there is no direct way to verify our results. Such verification was possible in our previous study of HD+ molecular ion,18 where the high-quality results of Tian22 and Pilon23 were available. Nevertheless, a less direct approach can be still used to evaluate the quality of our predictions. Since the oscillator strength decreases almost by an order of magnitude as ν increases by one (the ν = 0 and ν = 1 cases are an exception), the lifetime of a particular state is almost fully defined by the transition-dipole moments of not more than the three lowest transitions in Eq. (5). Table IV shows a comparison of the transition dipole moments of the ν(0 → ν ) transitions calculated in this work and the available experimental (Refs. 24 and 25) and theoretical BO (Refs. 26–29) results. As it can be seen, the present study yields results which are in a very good agreement with the experiment and with the most recent theoretical calculations. This indicates that the curve which shows the lifetime results presented in Figure 4 is accurate and reliable.
3
0.076 0.0742 0.0749 0.0613
IV. SUMMARY
In this work, a non-BO approach is used to study rovibrational states of the HD molecule corresponding to single rotational excitation (N = 1 states). The complete N = 1 rovibrational spectrum is calculated. The dissociation energies and the transition energies are also determined for N = 0 and N = 1 states and compared with recent results obtained with a method based on the BO approximation. The comparison shows a very good agreement of the two sets of results and indicates that the ECG basis set used in the present calculations can reliably describe states with low and high oscillatory behavior. This oscillatory behavior of the HD rovibrational states considered in this work is demonstrated using plots of the proton-deuteron correlation functions. In the present calculations, we also determined expectation values of the interparticle distances. These values are used to analyze the charge asymmetry in HD. The analysis shows that, as expected, the rotational excitation, elongates the average internuclear distance, but it has no significant effect on the charge asymmetry pattern of the N = 1 states as compared to the N = 0 states. Since according to the selection rule the N = 0 rovibrational states can only deexcite to the N = 1 states with smaller vibrational quantum numbers, the present calculation provides sufficient data for evaluating the interstate transition moments and lifetimes of the N = 0 states. The obtained lifetimes plotted as a function of vibrational quantum number show a similar behavior as those obtained for the HD+ ion in Ref. 18. ACKNOWLEDGMENTS
We are grateful to the University of Arizona Research Computing Center for the use of their computer resources. This work has been supported in part by the National Science Foundation (NSF) through the graduate research fellowship awarded to Keeper L. Sharkey; Grant No. DGE1-1143953. 1 S. Bubin, F. Leonarski, M. Stanke, and L. Adamowicz, J. Chem. Phys. 130,
124120 (2009). Cafiero and L. Adamowicz, Phys. Rev. Lett. 89, 073001 (2002). 3 Z. Lu, G. C. Tabisz, and L. Ulivi, Phys. Rev. A 47, 1159 (1993). 4 S. Bubin, M. Pavanello, W.-Ch. Tung, K. L. Skarkey, and L. Adamowicz, Chem. Rev. 113, 36 (2013). 5 J. Mitroy, S. Bubin, W. Horiuchi, Y. Suzuki, L. Adamowicz, W. Cencek, K. Szalewicz, J. Komasa, D. Blume, and K. Varga, Rev. Mod. Phys. 85, 693 (2013). 2 M.
FIG. 4. Lifetimes of the rotationless states.
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