Quantum dynamics study on the binding of a positron to vibrationally excited states of hydrogen cyanide molecule

Chemical Physics Letters 675 (2017) 118–123

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Research paper

Quantum dynamics study on the binding of a positron to vibrationally excited states of hydrogen cyanide molecule Toshiyuki Takayanagi a,⇑, Kento Suzuki a, Takahiko Yoshida a, Yukiumi Kita b, Masanori Tachikawa b a b

Department of Chemistry, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama City, Saitama 338-8570, Japan Department of Nanosystem Science, Graduate School of Nanoscience, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama, Kanagawa 236-0027, Japan

a r t i c l e

i n f o

Article history: Received 5 January 2017 In final form 8 March 2017 Available online 10 March 2017 Keywords: Positron Vibrationally elastic and inelastic cross sections Effective annihilation rates Positron affinity Hydrogen cyanide Molecular vibration

a b s t r a c t We present computational results of vibrationally enhanced positron annihilation in the e+ + HCN/DCN collisions within a local complex potential model. Vibrationally elastic and inelastic cross sections and effective annihilation rates were calculated by solving a time-dependent complex-potential Schrödinger equation under the ab initio potential energy surface for the positron attached HCN molecule, [HCN; e+], with multi-component configuration interaction level (Kita and Tachikawa, 2014). We discuss the effect of vibrational excitation on the positron affinities from the obtained vibrational resonance features. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Positrons have been widely used to probe negatively charged vacancy defects in bulk materials and also to detect cancer sites in human bodies using a positron-emission-tomography (PET) technique as an important medical application [1,2]. In spite of these widely-used applications of positrons, detailed interaction between positrons and molecules including positron binding mechanisms has not yet been fully understood from a viewpoint of molecular science. However, due to recent experimental developments to produce a high-resolution low-energy positron beam by Surko’s group [3–5], it is becoming uncovered that many molecules including organic compounds can bind positrons with the corresponding binding energies (or positron affinities) being in a range of a few meV to
300 meV depending on their molecular properties. In particular, it has been found that the magnitude of the measured positron binding energy roughly correlates with the molecular polarizability and permanent dipole moment of the molecule [6–9]. Notice that those positron binding energies are indirectly measured through the Feshbach-type resonance process, where a positron is transiently trapped to the molecule in the specific vibrationally excited state.

⇑ Corresponding author. E-mail address: [email protected] (T. Takayanagi). http://dx.doi.org/10.1016/j.cplett.2017.03.025 0009-2614/Ó 2017 Elsevier B.V. All rights reserved.

In order to understand detailed positron binding mechanisms in molecules at an atomic level, it should be highly important to compare experimental results with theoretical prediction. There have been some theoretical attempts to understand positron binding phenomena in molecules. The most important conclusion of the previous theoretical studies is that the electron-positron dynamical correlation is playing a crucial role for positron binding in molecules. Various methodologies including multi-component molecular orbital Hartree-Fock [10,11], many-body perturbation [12,13], configuration interaction [14–16], explicitly correlated Gaussian wave functions [17–19], and ab initio quantum Monte Carlo methods [20–23], have been so far applied. We should notice here that these first-principles ab initio calculations were carried out at the equilibrium structure under the fixed nuclei based on the Born-Oppenheimer approximation. Recently, the theoretical works of vibrational averaged positron binding is also reported under the multi-component ab initio potential energy surface for the positron attached molecule [24–29]. In addition to these elec tronic/positronic-structure-based studies, it should be mentioned that positron annihilation enhancement through the coupling between vibrational deformation of the molecular structure and virtual state formation was previously studied by Nishimura and Gianturco [30–32]; however, their study was based on a fixednuclei model and vibrational dynamics has not been explicitly taken into account.

T. Takayanagi et al. / Chemical Physics Letters 675 (2017) 118–123

In this paper, we report results of quantum scattering dynamics calculations for the e+ + HCN/DCN collision processes using a vertical positron affinity surface, which has been recently obtained as a function of the molecular geometry from the truncated multicomponent configuration interaction level of ab initio calculations [26]. According to the calculations, the vertical positron affinity value is highly dependent on the molecular geometry. Interestingly, it was found that vibrational excitation of the CN and CH stretching modes enhances the positron affinity compared to that at the HCN equilibrium structure. This conclusion is very important since, as mentioned above, positron binding energies have been experimentally measured through the vibrationally excited state Feshbach resonances of molecules, providing some cautions for determining accurate positron binding energies [26–29]. Since the HCN molecule has only three degrees of freedom, one can perform quantum dynamics calculations without reducing its dimensionality. Unfortunately, positron beam experiments have not yet been performed for HCN/DCN; however, we believe that the present dynamics calculations may provide a good theoretical model for understanding the effect of molecular vibration on positron binding. We here employ the local-complex-potential (LCP) model [33–42], which is based on a Born-Oppenheimer picture, to understand the dynamics of vibrational excitation and positron annihilation process. Within the simplest LCP model, the nuclear motions are governed by the complex-valued potential energy surface of V(Q)iC(Q)/2, where V and C are real and imaginary parts, respectively, and are functions of the nuclear coordinates Q. Notice that V and C respectively correspond to the positronic resonance energy position and its resonance width determining the positron capture/detachment lifetime at a given nuclear configuration, where these two quantities are assumed to be independent of positron collision energy. 2. Method The nuclear dynamics of resonant positron attachment within the LCP approximation can be described by the following timedependent Schrodinger equation (with  h = 1) as

i


 @wðR; r; h; tÞ i ¼ T N ðR; r; hÞ þ VðR; r; hÞ  CðR; r; hÞ wðR; r; h; tÞ @t 2 ð1Þ

where TN is the nuclear kinetic operator and (R, r, h) denote a set of the standard Jacobi coordinates for the H + CN configuration with R (= RHCN) and r (= rCN) being the radial coordinates and h being the angular coordinate. Here, the electronic/positronic potential energy surface is given by ViC/2, where V is the real part of the energy and C is the resonance width determining the positron capture/ detachment lifetime. All the calculations presented here were done with J = 0, where J is the total angular momentum of the triatomic HCN system. The time-dependent wave packet w is propagated in time with the following initial condition [38,40],

wðR; r; h; t ¼ 0Þ ¼ /v i ðR; r; hÞ ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðR; r; hÞ vv i ðR; r; hÞ 2p

ð2Þ

where vvi is the vibrational wavefunction of HCN in the vi-th vibrational state. The vibrationally elastic and inelastic cross sections can be calculated from the following equation as [40]

rv i ! v f ¼

4p 3 k

2

Z j

1 1

/v f ðR; r; hÞ expðiEtÞwðR; r; h; tÞdtj2

ð3Þ

where k is the wave vector corresponding to the initial positron collision energy. Similarly, the positron annihilation rate Zeff, which

119

can be expressed as an effective number of electrons per molecule, is written as [40]

Z eff ¼ 2pk

2

R1 1


2pk qd 2

wðR; r; h; tÞqd ðR; r; hÞ expðiEtÞwðR; r; h; t ¼ 0Þdt R1 wðR; r; h; tÞ expðiEtÞwðR; r; h; t ¼ 0Þdt 1

ð4Þ

where qd is the electron density at the positron position. In principle, the electron density should be parametrically dependent on the nuclear coordinates; however, we here assume that qd is a constant value. Thus, Zeff is simply determined by the Franck-Condon factor as described in Eq. (4). The positronic/electronic potential energy surface V in eq. (1) is written as

VðR; r; hÞ ¼ V n ðR; r; hÞ  EPA ðR; r; hÞ

ð5Þ

n

where V is the potential energy surface of the neutral HCN molecule. We have employed the potential energy surface developed by Bowman et al. on the basis of the ab initio CCSD(T)-level electronic structure calculations [43]. EPA is the vertical positron affinity surface and its functional form is simply approximated as 2 EPA ðR; r; hÞ ¼ EPA 0 þ a0 ðR  Re Þ þ a1 ðR  Re Þ

þ a2 ðr  r e Þ þ a3 ðr  r e Þ2 þ a4 ð1  cos hÞ

ð6Þ

þ a5 ðR  Re Þðr  r e Þ þ a6 ðR  Re Þð1  cos hÞ where EPA 0 (39.5 meV) is the positron affinity value at the HCN equilibrium structure (Re, re, h = 0). We [26] have previously calculated the PA values of HCN at a total of 2810 geometric grid points. The parameters a0a6 were determined from a least-square fitting to these calculated PA values (a0 = 42.855, a1 = 21.714, a2 = 16.789, a3 = 13.991, a4 = 96 942, a5 = 2.297, and a6 = 13.23). This fitting yielded the root-mean-square deviation and the maximum deviation to be 6.5 meV and 49.0 meV, respectively. As for the resonance width function, we here employ the simple form, CðR; r; hÞ ¼ a exp½bEPA ðR; r; hÞ; which is a function of the positron affinity value given in Eq. (6) in order to reduce the number of empirical parameters as much as possible. Although this functional form cannot describe the Wigner threshold law behavior appropriately, this form has been chosen so as that a positron can be effectively captured around the potential energy region where the positron affinity is small. In this work, a and b are chosen to be 50 meV and 0.08 meV1, respectively, in most of the present calculation. The parameter dependence on the computed results will be also given later. It should be mentioned that the above resonance width can be written as C = Cc + Ca, where Cc and Ca are the positron capture/detachment and annihilation widths, respectively. Theoretical work by Gribakin and Lee previously shows that the total resonance width C is exclusively determined by Cc due to the Cc >> Ca relation [44]. In order to numerically solve the time-dependent Schrodinger equation of Eq. (1), we have used the standard discrete-variable-r epresentation (DVR) grid basis functions for all the Jacobi internal coordinates [45]. The standard sine DVR functions were used for the R and r radial degrees of freedom, while the Legendre DVR function was used for the angular h coordinate. The wave packet was propagated by using the standard split-operator method for time-evolution, where the action of the exponential containing the radial kinetic energy operator was carried out using the fast-Fourier transform (FFT) algorithm [46–49]. The numerical parameters used in this study were carefully chosen from extensive convergence tests including vibrational energy states of HCN/DCN, and scattering cross sections and annihilation rates. The present calculations were done with the grid size of 128, 128, and 60 for the R [2–4.5 a0], r [1.8–3.8 a0], and h [0–180 deg.] coordinates (a0 is the Bohr radius), respectively. We have carried out the calculations with a larger size of 256, 256, and 80, and

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confirmed that the computed results are essentially the same. The time increment of the wave packet propagation was set to 5 atomic time unit (0.12 fs) and the total time step was chosen to be 40 000.

60

(a) 3

Figs. 1 and 2 show the potential energy surfaces for the neutral HCN and positronic HCN (denoted as [HCN; e+]), positron affinity surface, and resonance width surface. From the contour plots presented in Figs. 1b and 2b, the change in R and h significantly affects the magnitude of the positron affinity. In particular, it is seen that the positron affinity value decreases with an increase in the bending angle h. This suggests that vibrational excitation of HCN/DCN bending decrease the positron affinity. On the other hand, it is found that the positron affinity increases with an increase in R. This also suggests that vibrational excitation of R, i.e., CH stretching increases the positron affinity value in addition to vibrational anharmonic effects. These trends have already been discussed in our previous paper [26]. The results of the quantum scattering calculations for the e+ + HCN(0000)/DCN(0000) collision processes are presented in Figs. 3 and 4, where the vibrationally elastic cross sections for e+ + HCN(0000)/DCN(0000) ? e+ + HCN(0000)/DCN(0000) and effec-

20 3

2.5

3.0

30

(b)

0

40

5 10

20

50 40

20

0 2.0

2.5

3.0

R / a0

3.5

4.0

(c)

3.5

4.0

50

3 2

2.5

40

θ / deg.

1

2.0

20

40

1.5 2.0

2.5

3.0

3.5

R / a0

4.0

(b) 50 30

20

2.5

5

3.0

R / a0 Fig. 2. Potential energy surfaces of the HCN (black contours) and [HCN; e+] (red contours) (a), positron affinity surface (b) and width surface (c) plotted as a function of R and h with rCN being its equilibrium distance. Contour increment in panel (a) is 0.5 eV while that in (b) and (c) is 5 meV. The wavefunction of the HCN(0000) vibrational ground state is also shown in panel (a) with blue contours.

3.0

2.5

0 2.0

30

20 10

r / a0

4.0

60

4

(a)

40

2.0 5 0

1.5 2.0

10

2.5

3.0

R / a0

3.5

4.0

3.5

4.0

3.0

(c) 2.5

r / a0

3.5

R / a0

60

3.0

r / a0

2

0 2.0

θ / deg.

3. Results and discussion

θ / deg.

40

5 10

2.0 20 30

1.5 2.0

2.5

3.0

R / a0 Fig. 1. Potential energy surfaces of the HCN (black contours) and [HCN; e+] (red contours) (a), positron affinity surface (b) and width surface (c) plotted as a function of R and r for collinear configurations. Contour increment in panel (a) is 0.5 eV while that in (b) and (c) is 5 meV. The wavefunction of the HCN(0000) vibrational ground state is also shown in panel (a) with blue contours.

tive annihilation rates are plotted as a function of positron collision energy. The vibrationally elastic cross sections are plotted in unit of a20, while only relative annihilation rates corresponding to Zeff/qd are plotted due to the unknown (constant) qd parameter in eq. (4). Sharp resonances are seen to occur exactly at the same positions for both the vibrationally elastic cross sections and effective annihilation rates, as expected. Similar to the previous positron scattering experiments by Surko et al. [3–9], each resonance occurs slightly below the threshold energy of the HCN/DCN vibrational excitation and the corresponding energy shift is therefore providing the vibrational state specific positron affinity. The numerical data obtained from the present dynamics calculations are summarized in Table 1. The results presented in Table 1 indicate that bending excitation significantly decreases the positron affinity. This is simply because the PA value significantly decreases as the HCN/DCN molecule deforms from linear to bent structures due to the decrease in the dipole moment [26]. On the other hand, it is found that the excitation of both symmetric and antisymmetric stretching vibrations somewhat increases the positron affinity, where the antisymmetric stretching vibration (CH stretch) is more effective. This result also correlates with the PA value change along these stretching coordinates.

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0.20

Cross section / a0

0.15 0

(10 0)

0.10 0

0.05

(02 0) 0

(10 1)

0.00 0.0

0.2

0.4

0.6

0

(00 2)

0.8

(a)

0

(10 0)

0.15 0

(00 1)

0.10 0

(02 0)

0.05

0

0 (20 0)(10 1)

0.2

Positron collision energy / eV 2 1

0.6

0.8

1.0

2

10

(b)

+

0

e + HCN(00 0)

1

10

2

(b)

+

0

e + DCN(00 0)

2

0

10

0

0

-1

10

0

(10 0)

Cross section / a0

Cross section / a0

0.4

Positron collision energy / eV

10 10

0

0.20

0.00 0.0

1.0

+

e + DCN(00 0)

0.25

Annihilation rate / arb.

e + HCN(00 0)

2

0

2

(00 1)

+

Cross section / a0

0

Annihilation rate / arb.

0.30

(a)

(00 1)

(02 0)

-2

10

0

-3

(10 1)

10

0

-4

(04 0)

0

(00 2)

10

-5

10

-6

0

(10 0)

-1

10

0

(02 0)

0

(00 1)

-2

10

0

(20 0)

-3

10

0

(10 1)

-4

10

0

(04 0)

-5

10

0.0

0

10

0.2

0.4

0.6

0.8

10

1.0

0.0

Positron collision energy / eV Fig. 3. Vibrational elastic cross section (red curves) and annihilation parameter (blue curve) for the e+ + HCN(0000) collision obtained from the quantum dynamics calculations. Both linear scale (a) and logarithmic (b) plots are shown. The annihilation parameter is corresponding to Zeff/qd due to the unknown constant value of qd (see text for detail).

In Fig. 5 we compare the vibrational state resolved positron affinities obtained from the present dynamics calculations with the positron affinities averaged over vibrational wavefunction densities, which have been previously reported on the basis of the vibrational self-consistent field wavefunctions [26]. Although the present absolute values of the positron affinity are generally larger than the vibrational averaged values, good linear correlation is seen. This indicates that the vibrational averaged procedure can provide a general trend for the effect of vibrational excitation on the positron affinity value. Finally, we would like to give a brief comment on the resonance intensity observed in the calculated effective annihilation rates for HCN and DCN. In the case of e+ + HCN(0000) collision, the most intense enhancement is seen for the (0001) state, while the most intense enhancement is for the (1000) state in the DCN case. Within our theoretical framework, the resonance intensity can be exclusively determined by the feature of the positron capture/ detachment width function as well as PA surface, and cannot be explained by a simple analog of molecular infrared absorption spectra. In order to understand the dependence of the calculated effective annihilation rates on the positron capture/detachment width function, we have carried out the same scattering calculations but with three different parameter sets for (a, b) in the width function. Fig. 6 shows the computed results. It can be seen that the absolute values of the annihilation rate significantly depend on the parameters; however, it should be emphasized that resonance enhancement patterns via vibrational excitation are very similar. This suggests that the intensity ratio for different vibrationally excited states may not be sensitive to the choice of the width function. Although the width function used in this work is purely

0.2

0.4

0.6

0.8

1.0

Positron collision energy / eV Fig. 4. Vibrational elastic cross section (red curves) and annihilation parameter (blue curve) for the e+ + DCN(0000) collision obtained from the quantum dynamics calculations. Both linear scale (a) and logarithmic (b) plots are shown. The annihilation parameter is corresponding to Zeff/qd due to the unknown constant value of qd (see text for detail). Table 1 Resonance energies (Eres), vibrational threshold energies (Eth), and positron affinity values (EPA = Eth  Eres) obtained from the e+ + HCN(0000) and e+ + DCN(0000) scattering dynamics calculations. (m1m2lm3)a

Eres/eV

Eth/eVb

EPA/meV

HCN (0200) (1000) (0400) (0001) (2000) (0600) (0201) (1001) (0002)

0.137 0.215 0.314 0.365 0.469 0.486 0.539 0.619 0.758

0.177 0.261 0.349 0.414 0.520 0.518 0.583 0.672 0.817

40 46 35 49 51 34 44 53 59

DCN (0200) (1000) (0400) (0001) (1200) (0600) (0201) (2000) (1001) (0002)

0.103 0.194 0.244 0.283 0.340 0.382 0.424 0.428 0.515 0.602

0.142 0.239 0.281 0.329 0.384 0.416 0.465 0.476 0.565 0.652

41 45 37 46 44 36 41 48 51 50

a The linear triatomic convention is used for the vibrational states of HCN and DCN. m1, m2 and m3 are quantum numbers for symmetric stretch (CN stretch), bending, and asymmetric stretch (CH stretch), respectively. l is the projected angular momentum quantum number. b Threshold energy of vibrational excitation.

empirical in nature, it should be emphasized that different intensity patterns are obtained between HCN and DCN even if the same functional form is used for the capture width. More quantitative

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60 0

PA from Positron Scattering / meV

(00 2) 55 0

(10 1) 0

(20 0) 50

0

(00 1) 0

(10 0) 45

40

35

0

(02 0) 40

45

50

Vibrational averaged PA / meV Fig. 5. Correlation between the positron affinity obtained from the present e+ + HCN scattering calculations and the vibrational averaged positron affinity taken from Ref. [26].

Fig. 6. Effective annihilation rates (corresponding to Zeff/qd) as a function of positron collision energy calculated with different parameter sets for the width function.

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