H+ + NO(vi = 0) → H+ + NO(vf = 0–2) at ELab = 30 eV with canonical and Morse coherent states

Chemical Physics Letters 551 (2012) 42–49

Contents lists available at SciVerse ScienceDirect

Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett

H+ + NO(vi = 0) ? H+ + NO(vf = 0–2) at ELab = 30 eV with canonical and Morse coherent states Christopher Stopera, Buddhadev Maiti 1, Jorge A. Morales ⇑ Department of Chemistry and Biochemistry, Texas Tech University, P.O. Box 41061, Lubbock, TX 79409-1061, USA

a r t i c l e

i n f o

Article history: Received 11 July 2012 In final form 7 September 2012 Available online 15 September 2012

a b s t r a c t H+ + NO(vi = 0) = H+ + NO(vf = 0–2) at ELab = 30 eV is investigated with the simplest-level electron nuclear dynamics (SLEND) method. In a direct, time-dependent, variational, and non-adiabatic framework, SLEND adopts nuclear classical mechanics and an electronic single-determinantal wavefunction. A coherentstates (CS) procedure recovers quantum vibrational properties from classical mechanics. Besides canonical CS, SU(1,1), SU(2), and Gazeau–Klauder Morse CS are innovatively introduced to treat anharmonicity. SLEND vibrational differential cross, rainbow scattering angles, and H+ energy loss spectra compare well with experimental data and with vibrational close-coupling rotational infinite-order sudden approximation results obtained at a higher computational cost.  2012 Elsevier B.V. All rights reserved.

1. Introduction In 1979, Krutein and Linder conducted their seminal experiments on the H+ + M(vi = 0) ? H+ + M(vf) (M = N2, CO, and NO) reactions at ELab = 30, 50, and 78 eV [1], which are relevant in interstellar chemistry. With most measurements at ELab = 30 eV, Krutein and Linder reported vibrational energy transfers, H+ energy loss spectra [N2(vf = 0–1), CO/NO(vf = 0–2)], and vibrational differential cross sections [DCS; N2/NO(vf = 0), CO (vf = 0–2)] among other properties. While these results are undoubtedly important, theoretical studies of these experiments have been stymied by the inherent need of complete potential energy surfaces (PESs) in traditional dynamics methods. Thus, theoretical studies of these experiments have been reported only recently: Investigations of the three reactions with the vibrational close–coupling rotational infinite order sudden (VCC–RIOS) approximation [2–5], and another investigation of the H+ + N2 reaction with the quasi-classical trajectory (QCT) method [6]. Both methods employed computationally expensive PESs at the multi-reference configuration interaction (MRCI) level. In contrast to the previous methods, the simplest-level (SL) electron nuclear dynamics (END: SLEND) method [7] is a direct, time-dependent, variational and non-adiabatic method that requires no predetermined PESs to conduct simulations. Adopting classical mechanics for the nuclei and a single-determinantal wavefunction for the electrons, SLEND has described several proton-molecule reactions both feasibly and accurately [8–14]. ⇑ Corresponding author. Fax: +1 806 742 1289. E-mail address: [email protected] (J.A. Morales). Present address: Department of Chemistry, Georgia State University, Atlanta, GA 30302-4098, USA. 1

0009-2614/$ - see front matter  2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.09.016

Appropriately, for reactions involving state-to-state vibrational excitations, SLEND includes an intrinsic canonical coherent-states (CS) [15] procedure [8,9,16] to recover quantum vibrational properties from the SLEND classical nuclear dynamics. Successfully tested with H+ + H2(vi = 0) ? H+ + H2(vf = 0–5) at ELab = 30 eV [8,9], SLEND with its canonical CS procedure has been recently applied to the Krutein and Linder reactions for M = N2 and CO at ELab = 30 eV as a feasible alternative to the VCC–RIOS and QCT studies [17,18]. For H+ + N2, SLEND provided H+ energy loss spectra for vf = 0–1, vibrational energy transfers, and vibrational DCS for vf = 0 in very good experimental agreement and of comparable quality to their QCT [6] and VCC–RIOS [2] counterparts. Furthermore, SLEND predicted the best H+ + N2 scattering rainbow angle: Expt: R: R: R: hLab ¼ 9 ; hSLEND=6-31GR: ¼ 8:6 ; hVCC—RIOS ¼ 7 and hQCT ¼ 12 . Lab Lab Lab + For H + CO, SLEND provided vibrational DCS for vf = 0–2 in satisfactory experimental agreement but less accurate than its VCC–RIOS counterparts. However, SLEND again predicted the best R: R: H+ + CO scattering rainbow angle: hExpt: ¼ 9 ; hSLEND=pVDZ ¼ 9:18 Lab Lab VCC—RIOS R:  and hLab 7 . Following the H+ + N2 and H+ + CO studies [17,18], this investigation concentrates on H+ + NO at ELab = 30 eV as the concluding SLEND study of the Krutein and Linder experiments. As in previous cases, the H+ + NO SLEND results will not only be compared with Krutein and Linder’s measurements but also with the available VCC–RIOS results by Amaran and Kumar [5]. Moreover, a step forward into innovation is made in this investigation. As previously mentioned, the SLEND quantum vibrational recovery procedure [8,9,16] has hitherto involved the canonical CS set [15]: a mathematical construct that assumes the harmonic approximation for molecular vibrations. Since real molecules do not behave as harmonic oscillators, it would be more suitable to extend this vibrational recovery procedure to more accurate approximations.

43

C. Stopera et al. / Chemical Physics Letters 551 (2012) 42–49

Therefore, three different CS sets based on the Morse potential [19–21] are presently included in the CS analysis procedure. To the best of our knowledge, this is the first application of any Morse CS set in chemical dynamics. While the Morse potential is by no means the exact potential for molecular vibrations, it certainly constitutes a remarkable improvement over its harmonic counterpart. Since nothing is known of the behavior of any Morse CS with a real molecule, the present application focuses on the simplest vibrational case of a diatomic molecule. As such, NO is selected for this preliminary study over its experimental partners N2 and CO because of its higher anharmonic character. Applications to the remaining reactions in the Krutein and Linder experiments and to other molecules with higher anharmonic character will be published soon. This investigation is organized as follows: in Section 2, we discuss the basic aspects of SLEND and of its quantum vibrational analysis procedure with canonical and Morse CS sets. In Section 3, we discuss the computational details of the present SLEND simulations. In Section 4, we present and discuss the SLEND results for H+ + NO at ELab = 30 eV. Finally, in Section 5, we present some important concluding remarks.

b Total is the total Hamiltonian, the TDVP requires the where H Rt stationarity of the quantum action, dASLEND ¼ d ti2 LSLEND ½WSLEND ðtÞ; WSLEND ðtÞdt ¼ 0, with respect to all the variational parameters: fRi ðtÞ; Pi ðtÞ; zph ðtÞ; zph ðtÞg, and with proper boundary conditions. Then, the SLEND dynamical equations are [7,22,23]

2

iC

6 0 6 6 y 4 iCR 0

0

iCR

iC

iCR

iCTR

CRR

0

I

2 @E 3 32 dz 3 Total  dt 6 @E@z 7 7 6  dz 6 Total 7 7 6 0 7 76 dt 7 6 @z 7 7 dR 7 ¼ 6 @E 7 Total 7 I 56 4 dt 5 6 4 @R 5 dP @ETotal 0 dt 0

ð4Þ

@P

where ETotal is the total energy,

ETotal ½RðtÞ; PðtÞ; zðtÞ; zðtÞ  ¼

NN X P2 ðtÞ i

i¼1

þ

2M i

b e jzðtÞ; RðtÞi hzðtÞ; RðtÞj H hzðtÞ; RðtÞjzðtÞ; RðtÞi

ð5Þ

b e is the electronic Hamiltonian with nuclear repulsion included, H and

 @ 2 ln S  ;  @X ik @Y jl  0 R ¼R  @ 2 ln S  ; ðCXik Þph ¼   @zph @X ik  0  R ¼R @ 2 ln S  Cph;qg ¼   @zph @zqg  0

ðCXY Þik;jl ¼ 2Im 2. Theoretical background 2.1. The SLEND theory In-depth reviews of the END theory in its various forms have been presented elsewhere [7,22–24]. Thus, only a concise description of its simplest form, SLEND, is presented. END applies the time-dependent variational principle (TDVP) [25] to a trial total wavefunction in order to obtain dynamical equations. In SLEND, the trial total wavefunction, jWSLEND(t)i, is the product of a nuclear wavefunction and an electronic wavefunction, jR(t),P(t)i and jz(t),R(t)i, respectively. With NN nuclei, jR(t),P(t)i is the product of 3NN, 1-D, frozen, narrow, GAUSSIAN wave packets:

hXjRðtÞ; PðtÞi ¼

3N YN

(  ) 2 X i  Ri ðtÞ þ iP i ½X i  Ri ðtÞ ; 2DRi

exp 

i¼1

ð1Þ

with average positions {Ri(t)}, average momenta {Pi(t)}, and widths {DRi}. With Ne electrons and an atomic basis set of rank K > Ne, jz(t),R(t)i is a Thouless single-determinantal wavefunction [26] comprising Ne non-orthogonal, spin-unrestricted, dynamical spin– orbitals {vh} [7]:

hxjzðtÞ; RðtÞi ¼ detfvh ½xh ; zðtÞ; RðtÞg; ¼ wh ½x; RðtÞ þ

K X

vh ½x; zðtÞ; RðtÞ

wp ½x; RðtÞzph ð1 6 h 6 Ne Þ

ð2Þ

p¼N e þ1

The {vh} are linear combinations of Ne occupied, {wh}, and K–Ne virtual, {wp}, molecular spin–orbitals with the complex-valued Thouless parameters {zph(t)} as coefficients. The {wh,wp} are constructed at initial time at the time-independent unrestricted Hartree–Fock (UHF) level with the K atomic basis set functions centered on the wave packets’ centers {Ri(t)}. Due to the characteristics of the TDVP, the use of a Thouless single-determinantal state in SLEND affords various theoretical and numerical advantages [7]. The SLEND dynamical equations are obtained through the application of the TDVP to jWSLEND(t)i in the zero-width limit of the nuclear wave packets (DRi ? 0 "i). Starting with the quantum Lagrangian,

D LSLEND ½WSLEND ; WSLEND  ¼

 

 

b Total WSLEND WSLEND i @t@  H hWSLEND jWSLEND i

E ð3Þ

ð6Þ

R ¼R

are the dynamic metric matrices with S = hz0 (t), R0 (t)jz(t), R(t)i; CR and CRR are the SLEND non-adiabatic coupling terms. The SLEND Eqs. (4)–(6) express the coupled nuclear and electronic dynamics in a generalized symplectic form [25,27] through the conjugate variables {Ri(t),Pi(t)} and fzph ðtÞ; zph ðtÞg, respectively. While the electronic dynamics is quantum, the nuclear dynamics is rendered classical by the zero-width limit of the nuclear wave packets. 2.2. Vibrationally resolved properties via coherent states analysis The END method is intertwined with the CS theory [7,15]. Following Klauder’s initial definition [15], CS {jfii} are vectors in a Hilbert space that must satisfy two properties:  P1: The states {jfii} are continuous with respect to a set of parameters {fi}. ^ ¼ R dlðfi Þjfi ihfi j  P2: The states {jfii} attain resolution of unity 1 with a positive measure dl(fi) > 0. Sometimes, P2 is relaxed to admit CS sets with measures taking positive and negative values [21,28]. SLEND utilizes two CS sets [7]: Each 1-D GAUSSIAN wave packet jRi (t),Pi(t)i in Eq. (1) is a member of the canonical CS set with real parameters {fi} = {Ri,Pi} [15], whereas the Thouless single-determinantal state jz; Ri, Eq. (2), is a member of the Thouless CS set [15] with complex parameters {fi} = {zph}. Some CS sets are generated through the Perelomov prescription [29] that exploits the properties of Lie groups associated with particular Hamiltonians. The canonical and the Thouless CS sets are group-generated from the Heisenberg–Weyl and U(K) Lie groups associated with the 1-D b HO and electronic H b e Hamiltonians, respecharmonic-oscillator H tively. However, some CS sets are constructed through generalized prescriptions [30,31] unrelated to any Lie group. The original role of the canonical and the Thouless CS in SLEND was to provide a convenient parameterization for the TDVP treatment. However, the canonical CS set can play the additional role

44

C. Stopera et al. / Chemical Physics Letters 551 (2012) 42–49

Table 1 Morse ðJ; cÞ and dv ðnÞ, eigenstate probabilities, P Morse , and auxiliary equations (Aux. Eqs.) for three Morse coherent-states (CS) sets: SU(1,1) [19], SU(2) [20] and Coefficients, cMorse v v Gazeau–Klauder (GK) [21] as defined in Eq. (11). See text for further information. SU(1,1) CS jniSU(1,1)

Variables/Aux. Eqs. Morse

cv

SU(2) CS jn iSU(2)

GK CS jJ,ciGK nP

ðJ; cÞ

Morse

dv

ðnÞ

ð1  jnj2 Þ2k

h

Cðnþ2kÞ n!Cð2kÞ

i1=2

ð1 þ jnj2 Þ1v

nn

h i 2n ð1  jnj2 Þ2k Cn!ðnþ2kÞ Cð2kÞ jnj pffiffiffiffiffiffiffiffiffi 2mD ak jnj2 ¼ pffiffiffiffiffiffiffiffiffie

P Morse v 1st Aux. Eq.

2mDe þak

2nd Aux. Eq.

k ¼ 12 þ



Vib  2mE a2

ð1 þ jnj2 Þ1v

bN=2c Jm m¼0 qm

h

i1=2

h

i jnj2n

Cðv Þ n!Cðv nÞ

nP

Cðv Þ n!Cðv nÞ

½N=2 Jm m¼0 qm

jnj = exp{2ln[cos (jaj)]}  1

v¼

n

cen J 2p ei ffiffiffiffi

qn

nn

2

1=2

o1=2

8mDe a2

3rd Aux. Eq.

NA

sin2 ð2jajÞ þ v cos2 ð2jajÞ þ a8m 2 v EVib ¼ 0

4th Aux. Eq.

NA

NA

5th Aux. Eq.

NA

NA

Jn

qn

Eiv ib

qn ¼

Cðnþ1Þ CðNÞ ðNþ1Þ n CðNnÞ

xi 

qffiffiffiffiffiffiffiffiffi

o1

nP

N¼

x¼ en ¼

½N=2 J m m¼0 qm



8mDe a2



1=2

o1 P

½N=2 J j j¼0 qj

ej ¼ 0

1

1=2

2a2 De m n Nþ1 ðN

 nÞ

Table 2 Vibrational frequency, ve, anharmonicity constant, xeve, and anharmonictiy ratio, (xeve/ve)  100, 1 of N2, CO, and NO from experiments (Expt.) [39] and from SLEND/cc-pVDZ dynamical determinations. Molecule

ve (cm1) Expt./SLEND

xeve (cm1) Expt./SLEND

xeve  100/ve (%) Expt./SLEND

N2 CO NO

2358.57/2763.9 2169.81/2438.6 1904.20/2034.4

14.32/12.01 13.29/12.26 14.07/12.43

0.607%/0.435% 0.612%/0.503% 0.738%/0.611%

of recovering some quantum vibrational properties vanished by the zero-width limit of the nuclear wave packets [16]. Inspired by the distribution of exact classical energy transfer method by Giese and Gentry [32], that recovery procedure [16] exploits addib HO . Let m, x, and tional properties of the canonical CS set with H jviHO (v = 0,1,2 . . . ) be the mass, angular frequency and eigenstates b HO , then the normalized canonical CS admits both real jR,PiCCS of H and complex jniCCS parameterizations in its resolution into the jviHO

jR; PiCCS ¼

X

v ¼0;1...

cCCS v ðR; PÞjv iHO

¼ jniCCS X CCS ¼ dv ðnÞjv iHO ;

ð7Þ

v ¼0;1...

n ¼ ðmxÞ1=2 R2  iðmxÞ1=2 P2 with coefficients

  1 1 2 2 ½m cCCS ðR; PÞ ¼ exp  x R þ ðm x Þ P  v 4 ½ðmxÞ1=2 R  iðmxÞ1=2 P2 v pffiffiffiffiffi ;  v!   v 1 n CCS dv ðnÞ ¼ exp  jnj2 pffiffiffiffiffi : 2 v!

PCCS v

 P3: Temporal stability, the CS jR,PiCCS remains as a member of its own set during evolution: [25,27], jRðtÞ; PðtÞiCCS ¼ CCS b expðit H HO ÞjR; Pi . P3 implies that the normalized canonical CS

mx 1=4

p (

X  RðtÞ  exp  2DR

P3–P5 also hold under a TDVP evolution. The properties P3–P5 reveal that the quantum state hXjR(t), P(t)iCCS behaves as similarly to a classical harmonic oscillator as is possible with quantum mechanics. These properties lay the foundations of the SLEND vibrational recovery procedure. Consider a diatomic molecule AB simulated with SLEND. If AB is assumed harmonic, at any time of its SLEND evolution, it is possible to construct a canonical CS hXjR(t),P(t)iCCS with m = mAB = mAmB/(mA + mB) and x = xAB, whose quasi-classical vibration matches exactly the SLEND classical vibration of AB. Then, following Eqs. (7) and (8), the probability P CCS v of finding the SLEND-simulated molecule AB in the eigenstate jviHO is:

ð8Þ

jR,PiCCS is equal to any of the normalized wave packets jRi,Pii, Eq. (1), if R = Ri, P = Pi. b HO , the canonical CS exhibits additional When evolving with H properties [33]:

hXjRðtÞ; PðtÞiCCS ¼

with DR = (2mx)1/2, remains as a GAUSSIAN wave packet during evolution. b jR; PiCCS ¼ R and  P4: The CS average positions hR; PjCCS X CCS b CCS momenta hR; Pj P x jR; Pi ¼ P evolve as the positions and momenta of its classical-mechanics analogue with Hamiltonian HHO(R,P). P4 has been called the quasi-classical property [8,9,16–18].  P5: Minimum uncertainty relationship, the canonical CS satisfies DXDPX = 1/2 at all times.

)

2 þ iPðtÞ½X  RðtÞ

ð9Þ

 v  EAB =xAB Vib AB ¼ jcCCS v ðQ ; PÞj ¼ exp EVib =xAB v! 2v jnj CCS ¼ jdv ðnÞj2 ¼ expðjnj2 Þ v! 2

ð10Þ

AB 2 where PCCS v is a Poisson distribution in the variable EVib =xAB (or jnj ), 2 2 1 1 2 and EAB Vib ¼ 2mAB P AB þ 2 mAB xAB RAB is the classical vibrational energy of

the SLEND-simulated molecule AB. With PCCS v (and/or with its probCCS

ability amplitudes cCCS v ðR; PÞ or dv ðnÞÞ, quantum state-to-state vibrational properties can be calculated as shown in Section 4.2. In H+ + AB (vi) ? H+ + AB (vf) collisions, those properties’ calculations require applying the CS analysis procedure to AB only at the initial and final times. At those times, AB is completely separated from H+ and the aforementioned procedure is applicable to the isolated AB. The canonical CS vibrational recovery procedure for a molecule AB can be derived more rigorously from the original SLEND total

45

C. Stopera et al. / Chemical Physics Letters 551 (2012) 42–49 Morse

Table 1 lists the coefficients cMorse ðJ; cÞ; dv ðnÞ; eigenstate probav Morse 2 2 Morse Morse ¼ jcv ðJ; cÞj ¼ jdv ðnÞj ; and auxiliary equations bilities P v of the three Morse CS sets employed herein. Two of them were group-generated with the SU(1,1) {jniSU(1,1)} [19] and SU(2) {jniSU(2)} [20] Lie groups, respectively. Other Morse CS sets were developed through generalized prescriptions unrelated to any Lie group. For instance, the Gazeau–Klauder prescription [30] generates CS sets {jJ,ci} having real-valued, action-angle parameters: J > 0 and c,  1 < c < + 1; {jJ,ci} satisfy the original CS properties P1 and P2 as well as P3 and: b ci ¼ f ðJÞ, where H b is a given Hamil P6: Action identity, hJ; cj HjJ; tonian and f(J) a function of J alone. Originally [30], f(J) = xJ, where x is the angular velocity associated to c, but other functions are possible [21,36].

Figure 1. H+ + NO reactants initial conditions; spheres represent classical nuclei with projectile impact parameter b and target orientation [a,b].

AB AB wavefunction jWAB SLEND i ¼ jWe ijWN i [16,28]. By assuming the rotational-vibrational decoupling and the rigid-rotor approximations

[34], the nuclear wavefunction jWAB N i can be factored into AB AB translational jWAB Trans i, rotational jWRot i, and vibrational jWVib i waveAB AB AB functions: jWAB N i ¼ jWTrans ijWRot ijWVib i, which describe the centerof-mass translation, rotation, and vibration of AB, respectively.

Concurrently,

the

SLEND

total

energy

EAB Total

becomes

AB AB AB ABEq: AB AB , where EAB EAB Total ¼ ETrans þ EVib þ ERot þ Ee Trans ; ERot and EVib are the

is the energies of their corresponding wavefunctions and EABEq: e total electronic energy and nuclear repulsion at the AB equilibrium ABEq: Eq: ¼ EAB bond distance REq: e ðRAB Þ [16,28]. If AB is assumed AB ; Ee 2 1 EAB Vib ¼ 2mAB P AB þ V HO ðRAB Þ;V HO ðRAB Þ ¼

harmonic,

ABEq: EAB e ðRAB Þ  Ee

mAB x2AB 2

2 ðRAB  REq: AB Þ ¼

AB Vib i

and jW is a canonical CS describing the harmonic vibration of AB. In a SLEND simulation, the zero-width limit of the nuclear wave packets is taken. That makes the components of jWAB N i shrink into Dirac delta functions representing nuclei that obey classical mechanics. However, at any time of the SLEND evolution, it is possible to construct the canonical CS jWAB Vib i that matches the SLEND classical harmonic vibration of AB and then proceed with the vibrational recovery procedure as explained above. In some cases, the harmonic potential VHO(R) poorly models the vibration of a molecule and should be substituted for the more accurate Morse potential VMorse(R) = De{1  exp[a(R–REq.)]}2 with parameters a > 0, De > 0, and REq.. Thus, if AB is assumed Morse-like, 2 AB ABEq: 1 EAB (cf. FigVib ¼ 2mAB P AB þ V Morse ðRAB Þ; V Morse ðRAB Þ ¼ Ee ðRAB Þ  Ee ure 2), and jWAB i is a CS associated with the Morse Hamiltonian Vib b Morse . Various Morse CS sets [19–21,35,36] have been developed H and a succinct review of them follows (A comprehensive review will be published soon). Like the canonical CS set, the Morse CS sets admit real {jJ,ciMorse} and complex parameterizations {jniMorse} in b Morse eigenstates jviMorse(v = 0,1,2 . . . ): their resolutions into the H

jJ; ciMorse ¼

X

v ¼0;1...

jniMorse ¼

X

v ¼0;1...

cMorse ðJ; cÞjv iMorse ; v

Morse

dv

ðnÞjv iMorse

ð11Þ

P6 aims at expressing the CS average energy in a quasi-classical action-angle form. Notice that jniCCS is a Gazeau–Klauder CS with n = J1/2exp (ic) [30]. Roy and Roy [36] utilized the Gazeau–Klauder prescription to generate a Morse CS set that was further improved by Popov [21]. The latter set is hereafter denoted as the Gazeau– Klauder (GK) Morse CS set {jJ,ciGK} and is listed in Table 1. Another generalized prescription for CS was developed by Nieto and Simmons [31]: it provides minimum-uncertainty CS (MUCS) sets that satisfy the original CS properties P1 and P2 and also P5. Following this prescription, Nieto and Simmons constructed the MUCS Morse CS set [35]. The proliferation of Morse CS sets is in contrast with the sole canonical CS set for the harmonic case. This is so because the Perelomov, Gazeau–Klauder, and Nieto-Simmons prescriptions generate three different CS sets with VMorse(R) but the same canonical CS set with VHO(R). Additionally, with the Perelomov prescripb Morse tion, two Lie groups, SU(1,1) and SU(2), were associated with H but only one Lie group, the Heisenberg–Weyl one, was associated b HO . Despite their differences, all Morse CS sets transform with H into the canonical one in the limit of harmonic behavior. In regard b Morse , only the GK and MUCS to dynamics, when evolving with H Morse CS exactly satisfy properties P3 and P5, respectively. Nonetheless, the dynamics of a Morse CS can be unequivocally, though approximately, matched with the classical vibration of a Morselike molecule AB through their common potentials VMorse(RAB) and vibrational energies. Matching a Morse CS dynamics with the SLEND vibration of a Morse-like molecule AB involves obtaining the values of EAB Vib ; xAB ; a and De from the SLEND simulation (See Section 4.2). With those variables, the Morse CS probability PMorse v Morse (and/or its probability amplitude cMorse ðR; PÞ or dv ðnÞÞ is obv tained through the auxiliary equations listed in Table 1. Thereafter, the calculation of quantum vibrational state-to-state properties proceeds as with the canonical CS set case (see Section 4.2). The current SLEND/CS procedures are valid when the final target evolves into a single electronic state so that a definite harmonic/Morse potential can be fitted to the final vibration. That happened with H+ + N2 and H+ + CO [17,18], where the targets ended up in their ground electronic states due to large energy differences between ground and first-excited electronic states. In H+ + NO, SLEND predicts an average projectile-target charge-transfer probability of 14% (See Section 4.1) that reveals a small contribution of charge-transfer electronic states to the final target. However, with so small charge transfers, a final NO can be assumed in its electronic ground state and tractable with the current SLEND/ CS procedures (cf. the similar H+ + H2 case [8,9]). Systems undergoing vibrational excitations along with large charge transfers are not tractable this way but require high-level END/CS treatments. In addition to the vibrational CS sets, theoretical work has been invested into rotational and electronic CS. For instance, jWAB Rot i can be associated with a rotational CS [28]. Furthermore,

46

C. Stopera et al. / Chemical Physics Letters 551 (2012) 42–49

tional CS sets requires setting NO at rest (PN = PO = 0) in its groundstate UHF/cc-pVDZ equilibrium geometry. This is so because PCCS v and PMorse ¼ d0v when ENO v Vib ¼ 0 (cf. Eq. (10) and Table 1) so that no NO zero-point energy should be added to ENO Vib to reproduce jvi = 0i in a CS approach [9,16,32]; a NO target with ENO Vib > 0 admits a CS resolution into all the vibrational eigenstates jv P 0i. The angular orientation of NO, [a,b], is defined by the polar a (0 6 a 6 180) and azimuthal b(0 6 b 6 360) angles from its N terminus. The H+ projectile is initially located at the position (b, 0.0 a.u., 20.0 a.u.), where b P 0 is the impact parameter. The H+ initial momentum is PHþ ¼ ð0:0; 0:0; þ63:648633 a:u:Þ and corresponds to ELab = 30 eV. Due to the C1v symmetry of NO, independently varying a and b from 0 to 180 in steps Da,Db = 45 results in 17 unique initial orientations. For each of these orientations, b is varied in the range 0.0 a.u. 6 b 6 8.3 a.u. in steps Db = 0.1 a.u. The whole procedure generates 1428 initial conditions for the simulations. Each simulation is run for 2887 a.u. of time (=69.83 fs) to ensure a complete projectile-target separation after the collision. 4. Results and discussion 4.1. Reactivity Figure 2. Post-collision N–O nuclear relative distance vs. time of a H+ + NO at ELab = 30 eV simulation with initial target orientation [90, 0] and impact parameter b = 2.10 a.u. Results from SLEND/cc-pVDZ, classical harmonic oscillator (HO), and classical Morse oscillator. Potentials in the last two methods are best fits to the N–O SLEND/cc-pVDZ potential.

a quasi-classical electronic CS set has been suggested in the context of charge-equilibration models [37].

3. Computational details The present SLEND simulations have been performed with the CSDYN 1.0 code (A. Perera, T.V. Grimes and J.A. Morales, CSDYN 1.0, Texas Tech University, Lubbock, Texas, 2008–2010). The nuclear initial conditions for the H+ + NO SLEND calculations are shown in Figure 1. The initial electronic state of the target NO molecule, jWNO e i, is at the ground-state UHF level calculated with the cc-pVDZ atomic basis. The NO molecule is located with its center of mass at the origin of the laboratory frame. Reproduction of the initial vibrational eigenstate jWNO Vib i ¼ jv i ¼ 0i with all the discussed vibra-

SLEND/cc-pVDZ simulations of H+ + NO at ELab = 30 eV predict the following processes: (1) Scattering processes (SP)–nonreactive inelastic and elastic scattering of H+ from NO:

Hþ þ NOðv i ¼ 0Þ ! Hþ þ NOðv f Þ

ð12Þ

These processes primarily result in vibrational excitations, although rotational and electronic excitations occur to a lesser extent. (2) Collision induced dissociation (CID)–reactive scattering wherein NO dissociates:

Hþ þ NO ! Hq1 þ Nq2 þ Oq3 ;

q1 þ q2 þ q3 ¼ þ1

ð13Þ

CID only occurs in 1.8% of all the simulations. The lack of CID in H+ + NO follows a similar trend observed with SLEND in H+ + N2 and H+ + CO [17,18]. In SLEND, a simulation leading to SP describes simultaneously non-charge-transfer (NCT) H+ + NO ? H+ + NO and charge-transfer (CT) H+ + NO ? H + NO+ channels. The probabilities PNCT and PCT of

Figure 3. Vibrational differential cross section (DCS) for H+ + NO (vi = 0) ? H+ + NO(vf = 0–2) at ELab = 30 eV vs. scattering angle hLab. Results from SLEND/cc-pVDZ with the canonical [15], SU(1,1) [19], SU(2) [20] and GK [21] coherent states (CS), from VCC–RIOS [5] calculations, and from experiments [1].

47

C. Stopera et al. / Chemical Physics Letters 551 (2012) 42–49

Figure 4. H+ energy loss spectra computed at hLab = 5. Results from SLEND/cc-pVDZ with the canonical [15], SU(1,1) [19], SU(2) [20] and GK [21] coherent states (CS), from VCC–RIOS [5] calculations, and from experiments [1].

these processes are obtained by projecting the final SLEND electronic wavefunction jWei on all possible NCT and CT eigenstates, respectively [9,38]. While H+ + N2 and H+ + CO at ELab = 30 eV show negligible PCT  0% [17,18], H+ + NO at ELab = 30 eV exhibits an average PCT = 14%. This is sufficient to require the incorporation of PCT into the calculation of the vibrationally resolved properties as in the case of H+ + H2 at ELab = 30 eV [9]. 4.2. Vibrationally resolved properties with canonical and Morse CS sets Since properties calculated with the canonical and Morse CS sets will be compared, it is appropriate to first gauge the degree of anharmonicity of the molecules in the Krutein and Linder experiments. Consequently, Table 2 lists the frequency, ve, anharmonic constant, xeve, and anharmonicity ratio (xeve/ve)  100 of N2, CO, and NO obtained dynamically with SLEND/cc-pVDZ and from experiments [39]. The first approach involves the SLEND/cc-pVDZ simulations of the vibrations of the above molecules followed by a dynamical analysis to determine their frequencies ve and bestfitted Morse potentials VMorse(R). Specifically, an algorithm determines what analytical solution of the classical Morse oscillator fits the SLEND vibration the best. Having that best-fitted analytical solution, ve(=xAB/2p),xeve, and the VMorse(R) parameters a, REq. and D0 are readily determined/calculated. The same dynamical analysis is applied to each H+ + NO simulation below (cf. Figure 2). Molecules having anharmonicity ratios (xeve/ve)  100 > 10% are considered highly anharmonic (cf. the National Institute of Standards and Technology dataset). While N2, CO, and NO are not in that category, they do exhibit a moderate degree of anharmonicity whose hitherto unknown effect upon the vibrational CS procedure will be tested for NO in the next paragraphs. Notice that NO is the most anharmonic molecule in the Linder and Krutein experiments; that fact justifies the utilization of the Morse CS sets for NO first. The (xeve/ve)  100 ratio is not the only variable to gauge anharmonicity since dynamical aspects should be considered as well. It is well known that the harmonic approximation is adequate with low-energy molecular vibrations. Should a molecule be sufficiently excited in its vibration by a H+ collision, the harmonic approximation will break down. While NO is not excessively excited in collisions from intermediate and large b’s, NO displays an appreciable departure from harmonic behavior in collisions from small b’s, where vibrational energy transfers are large. Figure 2 illustrates that fact by comparing the NO SLEND/cc-pVDZ post-collision

vibration from the initial conditions [90, 0]/b = 2.10 a.u. with classical-mechanics vibrations with the harmonic VHO(R) and Morse VMorse(R) potentials that best fit the SLEND/cc-pVDZ potential energy. Figure 2 demonstrates that VMorse (R) reproduces exactly the SLEND/cc-pVDZ vibration under the examined conditions, a fact that calls for the utilization of Morse CS sets. An important measurement from the Krutein and Linder experiment is the H+ + NO(vi = 0) ? H+ + NO(vf = 0) DCS. Usual DCS expressions adopt the center-of-mass (COM) frame [40]. Therefore, the present DCS evaluations involve intermediate calculations in the COM frame. Partial wave analysis prescribes the COM DCS drv i ¼0!v f ðhCOM Þ=dX for the H+ + AB (vi = 0) ? H+ + AB (vf) NCT process from the target orientation [a,b] as [40]:

drv i ¼0!v f ðhCOM Þ dX v

fj f ðhCOM Þ ¼

2   X vf   ¼  fj ðhCOM Þ ;   j

1 X fð2l þ 1Þ: 2iki l

ð14Þ

 ½Prv f ðlÞ expð2igl Þ  d0v   Pl ðcos hCOM Þg v

where hCOM is the scattering angle, fj f ðhCOM Þ the scattering amplitude, ki the reduced incident momentum, l an orbital angular momentum quantum number (l  kib),gl the phase shift dgl(l)/ dl = hCOM(l)/2], and Pl(coshCOM) a Legendre polynomial of order l. Additionally, Prv f ðlÞ is the product of the NCT probability PNCT(l) (Section 4.1) and the CS vibrational excitation probability PCCS v f ðlÞ (Eq. (10)) or PMorse ðlÞ (Table 1). All the variables required in Eq. vf (14) are readily obtained from the SLEND data. The COM DCS are transformed back to the SLEND laboratory frame and averaged over all the target orientations [a,b] to obtain the final DCS. Figure 3 shows the SLEND/cc-pVDZ DCS for H+ + NO (vi = 0) ? H+ + NO (vf = 0–2) utilizing the canonical, SU(1,1), SU(2), and GK CS vs. the scattering angle hLab. For comparison, Figure 3 also includes the VCC–RIOS (vf = 0–2) [5] and the experimental (vf = 0) [1] DCS. Since the latter were reported in arbitrary units, they have been normalized so that their value at the experimental rainbow  R: scattering angle hExpt: ¼ 9 coincides with the SLEND/cc-pVDZ/ Lab GK DCS value at hLab = 9. For vf = 0, the RMS relative deviations of the five theoretical DCS from the experimental ones are: 0.2896 [SLEND/cc-pVDZ/Canonical], 0.2899 [SLEND/cc-pVDZ/SU (1,1)], 0.2886 [SLEND/cc-pVDZ/SU(2)], 0.2915 [SLEND/cc-pVDZ/

48

C. Stopera et al. / Chemical Physics Letters 551 (2012) 42–49

GK] and 0.5871 [VCC–RIOS], respectively. The four SLEND/cc-pVDZ DCS agree better with the experimental results than the VCC-RIOS DCS, especially at large hLab; SLEND/cc-pVDZ/SU(2) provides the best experimental agreement. The scattering rainbow angles from R: R: the DCS in Figure 2 are: hSLEND=cc-pVDZ ¼ 8 (All CS), hVCC—RIOS ¼ Lab Lab Expt: R: +   11:5 and hLab ¼ 9 . As with H + N2 and H+ + CO [2,3,17,18], the H+ + NO SLEND rainbow angle agrees significantly better with the experimental result than its VCC–RIOS counterpart. The discrepancies between the SLEND/cc-pVDZ and VCC-RIOS DCS become larger as vf increases. For vf = 0–2, the four SLEND/cc-pVDZ DCS differ slightly among themselves, but the canonical CS DCS deviate more perceptibly from the Morse CS DCS at large hLab. This is consistent with the previous analysis of the NO anharmonicity: Large values of hLab correspond to collisions from small b’s that provoke high vibrational excitations into the anharmonic regimen. A major result from the Krutein and Linder experiment is the H+ energy loss spectrum for H+ + NO (vi = 0) ? H+ + NO (vf = 0–2) that measures the mean vibrational excitation probabilities hP v i ¼0!v f ðhLab Þi: dr v

hPv i ¼0!v f ðhLab Þi ¼ P

i ¼0!v f

ground-state and first-excited-state PESs at the expensive MRCI/ cc-pVTZ level. This investigation demonstrates the possibility for the first time of utilizing Morse CS sets for the recovery of quantum vibrational properties. While NO exhibits some anharmonic features in the investigated collisions, they are not prevalent enough to make the SLEND/cc-pVDZ results with the three tested Morse CS sets markedly different from their canonical CS set counterparts. The best SLEND/cc-pVDZ DCS for vf = 0 was achieved with the SU(2) Morse CS set. Due to their complex mathematical nature, it is difficult to determine a priori which one of the three Morse CSs will provide the most accurate results and we have appealed to experiments to determine that. Due to its temporal stability, one could deem the GK Morse CS as the most accurate, but the present results do not support that conjecture. Future applications of this novel Morse CS vibrational recovery procedure will focus on the remaining reactions of the Krutein and Linder experiments, H+ + N2 and H+ + CO, and on systems exhibiting stronger anharmonic effects. Those studies will also involve additional Morse CS sets not tested herein: the MUCS Morse CS [35] and another GK Morse CS [36].

ðhLab Þ

dX drv ¼0!v ðhLab Þ 1 i f v f ¼0 dX

ð15Þ

Figure 4 compares the H+ energy loss spectrum at hLab = 5 from SLEND/cc-pVDZ with the four tested vibrational CS sets, from VCC–RIOS [5], and from the experiment [1]. The last result originally reported in arbitrary units has been normalized so that its vf = 0 peak has the same magnitude as the vf = 0 SLEND/cc-pVDZ/ GK peak. The RMS relative deviations of the five theoretical spectra from the experimental one are: 0.12220 (SLEND/cc-pVDZ/Canonical], 0.14473 [SLEND/cc-pVDZ/SU(1,1)], 0.12325 [SLEND/cc-pVDZ/ SU(2)], 0.16571 [SLEND/cc-pVDZ/GK] and 0.055501 (VCC–RIOS), respectively. All the theoretical spectra agree well with the experimental results, with the SLEND/cc-pVDZ spectra being only slightly inferior to their VCC–RIOS counterpart. The SLEND/cc-pVDZ spectra calculated with the Morse CS differ little from that with the canonical CS. This indicates that most of the vibrational energy transfers at hLab = 5 are too low to excite NO into the anharmonic regime. 5. Concluding remarks Completing a series of studies [17,18] of the reactions in the Krutein and Linder experiments [1], H+ + NO(vi = 0) ? H+ + NO(vf = 0–2) at ELab = 30 eV has been investigated with SLEND/ccpVDZ [7] in conjunction with a CS procedure to recover quantum state-to-state vibrational properties from the SLEND classical dynamics. In addition to the previously employed canonical CS set [17,18], three different Morse CS sets: SU(1,1) [19], SU(2) [20] and GK [21], were innovatively employed to take into account anharmonic effects. SLEND/cc-pVDZ with the four CS sets predicts   SLEND=cc-pVDZ DCS for vf = 0, rainbow scattering angles hLab ¼8; R: ¼ 9 , and H+ energy loss spectra for vf = 0–2 at hLab = 5 in hExpt: Lab very good experimental agreement. Moreover, the SLEND/cc-pVDZ DCS for vf = 0 and rainbow scattering angles exhibit better experimental agreements than their VCC–RIOS counterparts VCC—RIOS R:

¼ 11:5 , while the SLEND/cc-pVDZ H+ energy loss hLab spectra for vf = 0–2 at hLab = 5 are slightly inferior to their VCC–RIOS counterpart. Remarkably, SLEND/cc-pVDZ feasibly obtains its results through a classical-mechanics description for the nuclei and a single-determinantal representation for the electrons without employing predetermined high-level PESs. In contrast, VCC–RIOS [5] obtains its results through a demanding quantum treatment for the nuclei and by employing the predetermined

Acknowledgements This research is supported by the National Science Foundation Grants CHE-0645374 (CAREER) and CHE-0840493 (Instrumentation) and the Robert A. Welch Foundation Grant D-1539. References [1] J. Krutein, F. Linder, Journal of Chemical Physics 71 (1979) 599. [2] F.A. Gianturco, S. Kumar, T. Ritschel, R. Vetter, L. Zulicke, Journal of Chemical Physics 107 (1997) 6634. [3] T.J.D. Kumar, S. Kumar, Journal of Chemical Physics 121 (2004) 191. [4] T.J.D. Kumar, A. Saieswari, S. Kumar, Journal of Chemical Physics 124 (2006) 034314. [5] S. Amaran, S. Kumar, Journal of Chemical Physics 128 (2008) 124306. [6] T. Ritschel, S. Mahapatra, L. Zülicke, Chemical Physics 271 (2001) 155. [7] E. Deumens, A. Diz, R. Longo, Y. Öhrn, Reviews of Modern Physics 66 (1994) 917. [8] J.A. Morales, A.C. Diz, E. Deumens, Y. Öhrn, Chemical Physics Letters 233 (1995) 392. [9] J. Morales, A. Diz, E. Deumens, Y. Ohrn, Journal of Chemical Physics 103 (1995) 9968. [10] D. Jacquemin, J.A. Morales, E. Deumens, Y. Ohrn, Journal of Chemical Physics 107 (1997) 6146. [11] M. Hedström, J.A. Morales, E. Deumens, Y. Öhrn, Chemical Physics Letters 279 (1997) 241. [12] J.A. Morales, B. Maiti, Y. Yan, K. Tsereteli, J. Laraque, S. Addepalli, C. Myers, Chemical Physics Letters 414 (2005) 405. [13] B. Maiti, R. Sadeghi, A. Austin, J.A. Morales, Chemical Physics 340 (2007) 105. [14] B. Maiti, P.M. McLaurin, R. Sadeghi, S. Ajith Perera, J.A. Morales, International Journal of Quantum Chemistry 109 (2009) 3026. [15] J.R. Klauder, B.S. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics, World Scientific, Singapore, 1985. [16] J.A. Morales, Molecular Physics 108 (2010) 3199. [17] C. Stopera, B. Maiti, T.V. Grimes, P.M. McLaurin, J.A. Morales, Journal of Chemical Physics 134 (2011) 224308. [18] C. Stopera, B. Maiti, T.V. Grimes, P.M. McLaurin, J.A. Morales, Journal of Chemical Physics 136 (2012) 054304. [19] S. Kais, R.D. Levine, Physical Review A 41 (1990) 2301. [20] S.H. Dong, Canadian Journal of Physics 80 (2002) 129. [21] D. Popov, Physics Letters A 316 (2003) 369. [22] E. Deumens, Y. Ohrn, Faraday Transactions 93 (1997) 919. [23] E. Deumens, Y. Öhrn, Journal of Physical Chemistry A 105 (2001) 2660. [24] S.A. Perera, P.M. McLaurin, T.V. Grimes, J.A. Morales, Chemical Physics Letters 496 (2010) 188. [25] P. Kramer, M. Saraceno, Geometry of The Time-Dependent Variational Principle in Quantum Mechanics, Springer-Verlag, New York, 1981. [26] D.J. Thouless, Nuclear Physics 21 (1960) 225. [27] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1980. [28] J.A. Morales, E. Deumens, Y. Ohrn, Journal of Mathematical Physics 40 (1999) 766. [29] A. Perelomov, Generalized Coherent States and their Applications, SpringerVerlag, Berlin, 1986. [30] J.P. Gazeau, J.R. Klauder, Journal of Physics A: Mathematical and General 32 (1999) 123. [31] M.M. Nieto, L.M. Simmons Jr., Physical Review D 20 (1979) 1321.

C. Stopera et al. / Chemical Physics Letters 551 (2012) 42–49 [32] C.F. Giese, W.R. Gentry, Physical Review A 10 (1974) 2156. [33] C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics, Wiley-Interscience, New York, 1977. [34] F. Iachello, R.D. Levine, Algebraic Theory of Molecules, Oxford University Press, New York, 1995. [35] M.M. Nieto, L.M. Simmons Jr., Physical Review D 20 (1979) 1342. [36] B. Roy, P. Roy, Physics Letters A 296 (2002) 187. [37] J.A. Morales, Journal of Physical Chemistry A 113 (2009) 6004.

49

[38] P.M. McLaurin, New Applications of the Electron Nuclear Dynamics Theory to Scattering Processes and Chemical Reactions: Tool Development, Method Validation, and Computer Simulation, Texas Tech University, Lubbock, 2012. [39] K.P. Huber, G. Herzberg, Constants of diatomic molecules, in: P.J. Linstrom, W.G. Mallard (Eds.), NIST Chemistry WebBook, NIST Standard Reference Database Number 69, National Institute of Standards and Technology, Gaithersburg MD, 2011. [40] M.S. Child, Molecular Collision Theory, Academic Press, Inc., New York, 1974.