Ab Initio Potential Energy Curves for the Ground and Low Lying

Article pubs.acs.org/JPCA

Ab Initio Potential Energy Curves for the Ground and Low Lying Excited States of NH− and the Effect of 2Σ± States on Λ‑Doubling of the Ground State X2Π Saurabh Srivastava† and N. Sathyamurthy*,†,‡ †

Department of Chemistry, Indian Institute of Technology, Kanpur, Kanpur 208016, India Indian Institute of Science Education and Research, Mohali, Sector 81, SAS Nagar, Manauli, P.O. 140306, India

‡

S Supporting Information *

ABSTRACT: Complete basis set extrapolated ab initio potential energy curves obtained from multireference configuration interaction (MRCI) level calculations for the ground state (X2Π) and the a4Σ− state of NH− and the ground state (X3Σ−) of NH are reported. The potential energy curves for the A′2Σ− and A2Σ+ states of NH− have been computed using the V6Z basis set at the MRCI level. Λ-doubling parameters p and q are calculated for the ground and the first excited vibrational states of the ground electronic state of NH− using second-orderperturbation theory. The effect of the 2Σ+ and 2Σ− states on the Λ-doubling values is discussed. Earlier experiments had not considered the influence of the 2Σ− state on p and q while fitting the spectral data. Using the computed potential energy curves and the ro-vibrational spectra including the fine splitting, we have computed the threshold for electron detachment. The result is in agreement with the experimental values of Neumark et al. [J. Chem. Phys. 1985, 83, 4364] and Farley et al. [Phys. Rev. A 1987, 35, 1099].

■

by Neumark et al.17 They reported that the autodetachment rate was larger for the upper Λ-doublet level of NH−(v = 1) than for the lower level. They found that rotational-electronic coupling played an important role in explaining the differences in the Λ-doublet autodetachment rates. In another laser-ion beam experiment by Farley and co-workers,18−20 a high resolution ro-vibrational spectrum of NH− was recorded. Cade21 used Hartree−Fock method to obtain the electron affinity (EA) of some hydrogen containing diatomic species including NH. Rosmus and Meyer22 calculated the electron affinity and other spectroscopic constants for NH− and other diatomic anions using the wave function obtained from the coupled electron pair approximation (CEPA) based on pseudo natural orbital configuration interaction (PNO−CI) expansion. The CEPA wave functions yielded an electron affinity of 0.01 eV, indicating a marginal stability in contrast to the experimental value of 0.38 eV. Frenking and Koch23 used the Møller−Plesset perturbation theory up to fourth order (MP4) to calculate the adiabatic electron affinity for diatomic hydrides and reported a value of 0.18 eV for NH−. Mänz et al.24 calculated the potential energy, electric dipole, and electronic transition moment functions for several bound electronic states of NH− using multiconfiguration-self-consistent-field-configuration interaction

INTRODUCTION Molecular anions are known to be important in various phenomena occurring in the earthʼs ionosphere,1 plasmas, and interstellar media and in Titanʼs deep ionosphere.2 However, they are difficult to study theoretically. Even the simplest of them, H2−, poses a challenge in view of the neutral molecule plus free electron state resulting in what is called the variational collapse.3−6 Some of the other hydrogen containing anions like BH−, CH−, NH−, and OH−, formed by the second row atoms have been studied in the past experimentally7−10 and theoretically.11−13 Both CH− and OH− show photodetachment involving excited electronic states. NH− is known to exhibit autodetachment from its first excited vibrational state of the ground electronic state. The role of spin−orbit coupling including excited states and spin forbidden transitions has been studied recently14 at the multireference configuration interaction (MRCI) level of theory for the CH− anion. Out of all the above-mentioned anions, NH− poses the greatest challenge for theory. The first photoelectron spectroscopic experiment reported in 1974 by Celotta et al.15 showed one peak and yielded an electron affinity of 0.38 ± 0.03 eV, whereas Engelking and Lineberger16 observed two peaks, corresponding to the transition from the X2Π state of NH− to the X3Σ− and a1Δ states of NH. They16 also observed that the first excited (v = 1) vibrational state of the ground electronic state of NH− has sufficient energy for autodetachment. The first infrared spectrum of NH− was obtained through autodetachment spectroscopy using a coaxial laser-ion beam spectrometer © 2013 American Chemical Society

Special Issue: Structure and Dynamics: ESDMC, IACS-2013 Received: March 20, 2013 Revised: June 12, 2013 Published: June 13, 2013 8623

dx.doi.org/10.1021/jp4027628 | J. Phys. Chem. A 2013, 117, 8623−8631

The Journal of Physical Chemistry A

Article

(MCSCF-CI) method. They also reported the spectroscopic constants for the A2Σ+ state of NH− and transition probabilities for the X2Π → A2Σ+ transition. Gutsev and Bartlett25 reported the electron affinity for NH obtained using coupled-cluster and Hartree−Fock density functional theories. Feller and Sordo26 reported spectroscopic constants using the coupled cluster singles, doubles and triples (CCSDT) method for the first row diatomic hydrides and their anions including NH−. Wan et al.27 computed the effect of the spin−orbit coupling on the low lying states of NH−. Unfortunately, their results were not dependable because the potential energy curves did not have the correct asymptotic limit. Recently, Rajvanshi and Baluja28 used the R-matrix method to calculate the (integral and differential) elastic, momentum transfer, excitation, and ionization cross sections for the electron impact on the NH radical. They obtained a stable bound state for the X2Π state of NH−. In the present work we report the potential energy curves obtained using the multireference configuration interaction (MRCI) method for the ground and low lying excited states of NH− and the ground state of NH. A complete basis set (CBS) extrapolation has also been carried out for the ground state of NH and NH−. We have calculated the spin−orbit coupling and Λ-doubling for the ground state of NH− and have examined the effect of the excited states (2Σ+ and 2Σ−) over the Λ-doubling. The threshold for electron detachment from NH− is also calculated. The theoretical methods employed in our study are described briefly below. That is followed by the results and discussion of our findings and a summary and conclusions of our work.

+

⟨Λ , Σ , Ω, S , v|H SO|Λ , Σ , Ω, S , v⟩ = AΛ, v ΛΣ

HROT = (1/2μR2)R2 = (1/2μR2)(R x 2 + R y 2)

(5)

J≡R+L+S

(6)

HROT = (1/2μR2)[(J2 − Jz 2 ) + (L2 − Lz 2) + (S2 − Sz 2) + (L+S− + L−S+) − (J+ L− + J− L+) − (J+ S− + J− S+)]

(7)

where μ and R are the reduced mass and the internuclear distance, respectively, and R, L, S, and J are the nuclear rotational, orbital, spin, and total angular momenta, respectively. The raising and lowering operators are defined as J± = Jx ± iJy ,

(1)

L± = L x ± iL y ,

S± = Sx ± iSy

(8)

The last three terms of eq 7 have off-diagonal elements, which are treated as rotational perturbations. These perturbations are defined as spin-electronic homogeneous, L-uncoupling heterogeneous, and S-uncoupling heterogeneous perturbations, respectively. The selection rules for each of these perturbations are given in Table 1. It is evident from the selection rules that

where H0 and HSO are the unperturbed Hamiltonian and the spin−orbit coupling, respectively. The Breit−Pauli perturbation term (HSO) is given as29 ⎫ α2 1 Z α 2 ⎧ ZA Σ⎨ 3 li A ·si + B3 li B·si⎬ − Σ (rij × pi) 2 i ⎩ riA 2 i ≠ j rij 3 ri B ⎭ × (si + 2sj)

(4)

where S and v are spin angular momentum and vibrational quantum number, respectively. The calculation of off-diagonal matrix elements of HSO and the spin−orbit coupling constant for the X2Π state of NH− is discussed later in the text. Λ-Doubling. Apart from the spin−orbit perturbation, the off-diagonal terms neglected in the rotational part of the Hamiltonian give rise to rotational perturbations. The rotational Hamiltonian (HROT) can be written as29,31

THEORETICAL METHODS Spin−Orbit Coupling. The interaction between the spin and orbital angular momenta of the electrons is added as a perturbation to the electronic Hamiltonian and the total Hamiltonian operator is written as

H SO =

ΔΛ = −ΔΣ = 0, ± 1;

−

(3) Σ ↔Σ It should be noted here that, in the single-configuration limit, if the two interacting states belong to the same configuration, then ΔΛ = ΔΣ = 0. If the two states differ by at the most one spin−orbital, then ΔΛ = −ΔΣ = ±1. This selection rule allows spin−orbit coupling between X2Π, 2 − A′ Σ , and A2Σ+ states of NH−. The spin−orbit coupling constant (A) can be defined for the diagonal matrix elements of HSO such that29

■

HTOT = H 0 + H SO

ΔS = 0, ±1;

ΔΩ = 0;

Table 1. Perturbation Operators and Selection Rules

(2)

operator

where the index i refers to the electrons and A and B refer to the two nuclei in a diatomic species AB. α = 1/137.036 is the fine structure constant and l and s are the orbital and spin angular momenta, respectively. p is the linear momentum of the electron. riA and riB are the distances between the ith electron and the nuclei A and B, respectively, and rij is the distance between electrons. The first part of eq 2 is defined as the direct spin−orbit interaction and is a single electron operator, and the second part is the spin-other orbit interaction and a two electron operator. Due to the coupling of angular momenta, the total orbital angular momentum Λ and the total spin angular momentum Σ are no longer good quantum numbers. The good quantum number Ω = Λ + Σ and the states corresponding to Λ and Σ correlate with the components of Ω. The X2Π state of NH− splits into 2Π3/2 and 2Π1/2 components. The selection rules30 for the matrix elements of HSO can be summarized as

perturbation

selection rules

(1/2μR2)(L+S− + L−S+) spin-electronic (ΔΩ = 0, ΔS = 0, ΔΛ = −ΔΣ = ±1) (1/2μR2)(J+L− + J−L+) L-uncoupling (ΔΣ = 0, ΔS = 0, ΔΛ = ΔΩ = ±1) (1/2μR2)(J+S− + J−S+) S-uncoupling (ΔΛ = 0, ΔS = 0, ΔΩ = ΔΣ = ±1)

the spin-electronic homogeneous perturbation term and the L-uncoupling perturbation term involve interaction between two different electronic states, but the S-uncoupling perturbation mixes different components of the same (multiplet) electronic state. It has been observed that the S-uncoupling term is responsible for the evolution to Hundʼs coupling case b from case a with an increase in the total angular momentum quantum number J. Similarly, the evolution from Hundʼs case a to case d, with an increase in J, occurs due to the L-uncoupling term. When Λ is nonzero, the L-uncoupling operator allows interaction with other electronic states, thus lifting the degeneracy of 8624



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Figure 1. Potential energy curves for (a) X2Π, (b) A′2Σ−, (c) A2Σ+, and (d) a4Σ− of NH− obtained at the MRCI level of theory using AVXZ and VXZ (X = D, T, Q, 5, 6) basis sets. The CBS extrapolated potential energy curve for NH (X3Σ−) is included for comparison.

a given Λ state. This splitting of a Λ-state is termed as Λ-doubling. For example, a 2Π state will have Λ values of ±1, which are degenerate and an interaction with a nearby 2Σ± state would split the rotational levels corresponding to the two values of Λ. It is important to note that the two interacting states are sufficiently far apart in energy. The second-order perturbation theory is used to calculate the perturbation energies. The final matrix including all the perturbations discussed above, that is, spin−orbit, spin-electronic, L-uncoupling, and S-uncoupling for a 2Π state interacting with 2 ± Σ states is written as32

m12 = m21 ⎧⎛ 3 ⎞⎛ 1 ⎞⎫ = −⎨⎜J + ⎟⎜J − ⎟⎬ ⎝ ⎠ ⎝ ⎩ 2 2 ⎠⎭ ⎡ q ⎤⎡ − ( −1)(J − 1/2)⎢ v ⎥⎢J + ⎣ 2 ⎦⎣

γ⎞ ⎧⎛ ⎨⎜Bv − v ⎟ ⎝ ⎩ 2⎠ 1 ⎤⎫ ⎬ 2 ⎥⎦⎭

1/2

m33 =

Av 7 + Bv J(J + 1) − 2 4

{ {

} }

Av 1 + Bv J(J + 1) + 2 4 ⎡ pv ⎤⎡ 1⎤ × ⎢ + q v ⎥⎢J + ⎥ ⎣2 ⎦⎣ 2⎦

m44 = −

(12)

(13)

− γv + ( −1)(J − 1/2) (14)

m34 = m43 ⎧⎛ 3 ⎞⎛ 1 ⎞⎫ = −⎨⎜J + ⎟⎜J − ⎟⎬ ⎝ ⎠ ⎝ ⎩ 2 2 ⎠⎭ ⎡ q ⎤⎡ + ( −1)(J − 1/2)⎢ v ⎥⎢J + ⎣ 2 ⎦⎣

γ⎞ ⎧⎛ ⎨⎜Bv − v ⎟ ⎝ ⎩ 2⎠ 1 ⎤⎫ ⎬ 2 ⎥⎦⎭

1/2

where the matrix elements are given by m11 =

Av 7 + Bv J(J + 1) − 2 4

{

}

Av 1 + Bv J(J + 1) + 2 4 ⎡ pv ⎤⎡ 1⎤ × ⎢ + q v ⎥⎢J + ⎥ ⎣2 ⎦⎣ 2⎦

m22 = −

{

(10)

(15)

In these equations, Av is the spin−orbit coupling constant as defined in eq 4 and Bv is the rotational constant for a given vibrational state (v) and is written as ⟨v|(ℏ2/2μR2)|v⟩. The term γv arises from the interaction between the electron spin and nuclear rotation and it is taken to be zero in the present work.

} − γ − (−1)

(J − 1/2)

v

(11) 8625



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This is because γv is usually very small for small hydrogen containing species. For example, γv = −0.119 cm−1 for OH (X2Π) (see p 361 of ref 32). Mulliken and Christy33 defined the terms pv and qv, known as Λ-doubling constants, arising from the interaction between all the 2Σ states and the 2Π state. These terms are given by pvΠ (2 Σs) = 2

∑ (−1)s ⟨2Π, v|Σaili+si−|2Σs , v′⟩ i

2

Σ, v ′ 2 s

Σ , v′

×

ℏ2 Σl − 2μR2 i i

Π, v

=2

2

Π, v

E Π, v − E Σ, v 2

qvΠ(2 Σs)

to establish the correct ground and excited state potential energy curves of H2− and CH−, the same approach is used in the case of the excited states a4Σ−, A′2Σ−, and A2Σ+ of NH−. The second moments were calculated at the MRCI/VXZ and MRCI/AVXZ levels of theory for R in the range 0.8a0−10a0. The results obtained for the excited state A′2Σ− are plotted in Figure 2 in the form of the square root of ⟨r2⟩ and its x, y, and z components as a function of R. The plots for other states are given in Supporting Information as Figure SF1 and SF2. It is clear from the plots that the augmented basis sets allow the electron to be far away from the center of mass of the diatom. This results in an increase in the well depth in the potential energy curve in the Franck−Condon region. Therefore, the basis set cc-pVXZ (X = Q(4), 5, 6) has been used to obtain the CBS potential energy values for the a4Σ− state and then the V6Z basis set for the A′2Σ− and A2Σ+ states. The resulting potential energy curves for the ground and the excited states of NH− are plotted in Figure 3. It is to be noted here that the small undulations in Figure 1c and the top curve in Figure 3 could be attributed to avoided crossings and mixing of states. These curves were obtained with small basis sets and are higher in energy where other states of the same symmetry could arise. The different asymptotic atomic and ionic states are listed in Table 2. Vibrational bound state calculations have been performed using the CBS extrapolated ab initio potential energy curves for the ground state X2Π of NH− and the ground state X3Σ− of NH, whereas the V6Z potential energy curve was used for the A′2Σ− and A2Σ+ states of NH−. These calculations were performed using the Fortran code LEVEL 8.0 of LeRoy.40 The resulting spectroscopic constants for the ground and excited state of NH− and the ground state of NH are reported and compared with the earlier reported41,42 values in Table 3. It is important to point out that the first excited state obtained in this calculation for NH− is 2Σ− and not 2Σ+ as reported in earlier works.24 The reason for this difference could be the assumption that, because NH− is isoelectronic to OH, it must have a similar order of electronic states. The 2Σ− state is repulsive and higher in energy than the 2Σ+ state in the Franck−Condon region for OH. We find that both A′2Σ− and A2Σ+ states support bound states for NH−. This result has important implications on the Λ-doubling parameters (vide infra). Fine Structure and Λ Doubling. Neumark et al.17 and Al-Zaʼal et al.19 observed experimentally the fine structure including Λ-doubling for the ground electronic state of NH− for the vibrational states v = 0 and 1. They fitted their data to a model 2Π Hamiltonian with adjustable parameters developed by Zare et al.43,44 This model Hamiltonian was constructed by considering the excited state to be a 2Σ+ state influencing the rotational perturbation causing Λ-doubling in the ground state (X2Π) of NH−. We find that both the A′2Σ− and the A2Σ+ states contribute to the Λ-doubling parameters of the ground electronic state of NH−. The molecular orbital configuration for the X2Π, A′2Σ−, and 2 + A Σ states of NH− is σ21sσ22sσ22pπ32p, σ21sσ22sσ22pπ22pσ2p *1 and σ21sσ22sσ12pπ42p, 2 3 2 2 1 respectively. Briefly, they are denoted as σ π , σ π σ , and σ1π4. To obtain the fine structure and the Λ-doubling using the matrix elements in eq 9, we need to calculate the spin−orbit coupling constant and the contribution of the Λ-doubling parameters p and q from each of the A′2Σ− and A2Σ+ states. The z component of the spin−orbit coupling operator (ailziszi) has all the diagonal elements and for a σ2π3 configuration, ⟨2Π1/2|HSO|2Π1/2⟩ and ⟨2Π3/2|HSO|2Π3/2⟩ have values equal to −Ae and +Ae, respectively. It is, therefore, an inverted state when compared to

s

∑ (−1) 2

Σ, v ′

(16)

′

ℏ2 Σl + 2μR2 i i

2

2

s

Σ , v′

E Π, v − E Σ, v

′

(17)

where the summation goes over all the vibrational levels of all the relevant excited 2Σ± electronic states. The exponent s is zero for 2Σ+ states and 1 for 2Σ− states. The operators l+i and s‑i are the raising and lowering operators for the orbital angular momentum and the spin angular momentum, respectively, for the ith electron. The operator ai contains the radial part of the Breit−Pauli Hamiltonian. Because both the 2Σ+ and 2Σ− states contribute to the Λ-doubling parameters, we can write 2

pvΠ = pvΠ ( Σ+) + pvΠ (2 Σ−)

(18)

2

qvΠ = qvΠ( Σ+) + qvΠ(2 Σ−)

(19)

It is to be noted here that there is one more parameter (ov) defined by Mulliken and Christy but it is not included here because it does not make any contribution to the Λ-doubling of the 2Π state. The calculation of the Λ-doubling parameters pv and qv and the contribution of the 2Σ± (excited) states will be described later in the text.

■

RESULTS AND DISCUSSION Potential Energy Curves and Vibrational Bound States. Potential energy values are calculated as a function of the internuclear distance (R) of NH−, ranging from 0.8 a0 to 15.0 a0 at the MRCI level of theory using aug-cc-pVXZ(AVXZ) and cc-pVXZ(VXZ) basis sets34−36 and the MOLPRO37 suite of programs. State averaged full valence CASSCF calculations with four σ and two π orbitals were performed to obtain the reference orbitals for the MRCI calculation. The computed potential energy values were extrapolated to the complete basis set (CBS) limit using the uniform singlet and triplet electronpairs (USTE) method38,39 proposed by Varandas for the ground state of NH and the ground and the lowest excited states of NH−. A summary of this method is given in the Appendix. The potential energy curves obtained using the AVXZ and VXZ (X = D, T, Q, 5, 6) basis sets for the ground and excited states of NH− are plotted in Figure 1. The potential energy curve for the ground state of NH at the CBS limit is included in the same plot for quick reference and it is clear that the excited states of NH− are embedded in the continuum of the neutral molecule plus electron (Figure 1b−d. The use of a large basis set for the ground state of NH− results in a potential energy curve that is lower than that of the neutral NH (Figure 1a). As we have used6,14 the second moment calculations successfully 8626



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Figure 2. Values of (a) ⟨r2⟩1/2, (b) ⟨x2⟩1/2, (c) ⟨y2⟩1/2, and (d) ⟨z2⟩1/2 obtained at the MRCI level of theory using AVXZ and VXZ (X = D, T, Q, 5, 6) basis sets for the A′2Σ− state of NH−.

Table 2. Electronic States of NH− and NH and Their Dissociative Atomic and Ionic Components along with the Energies at an Internuclear Distance of 100a0 species

state

dissociative atomic and ionic states

energy (hartree)

NH−

X2Π A′2Σ− A2Σ+ a4Σ− X3Σ−

N−(3P) + H(2S) N−(3P) + H(2S) N−(1D) + H(2S) N(4S) + H−(1S) N(4S) + H(2S)

−55.00871780a −55.00208184b −54.95275350b −55.04089308a −55.02533333a

NH a

At the MRCI/CBS level of theory. bAt the MRCI/V6Z level of theory.

The off-diagonal elements of the spin−orbit matrix for a diatomic hydride can be calculated in terms of the spin−orbit coupling constant of the diagonal elements using a pure precession approximation. This approximation requires that each of the interacting 2Π and 2Σ states is well described by a single configuration and that the two states differ by a single spin−orbital. In addition, the spin−orbital is a pure atomic orbital such that29

Figure 3. Potential energy curve for the ground state (X2Π) of NH−, the excited state a4Σ− and the ground state (X3Σ−) of NH at the MRCI/CBS level of theory. The excited states A′2Σ− and A2Σ+ of NH− are calculated at the MRCI/V6Z level of theory.

l±|nlλ⟩ = [l(l + 1) − λ(λ ± 1)]1/2 |nlλ ± 1⟩

a π1 configuration. We have used the MOLPRO suite of programs37 to calculate the spin−orbit constant (Ae) for internuclear distance values in the range 0.8a0−15.0a0, and the results are plotted in Figure 4. The spin−orbit constant (Av) for a given vibrational level is calculated as Av =

∫R |χv (R)|2 Ae(R)

(21)

where n, l, and λ are quantum numbers for an atomic orbital. The approximation takes the simple pure precession form for the X2Π−A2Σ+ interaction because the valence orbital configurations corresponding to these two states can be written as π1 and σ1, respectively, as the other orbitals are completely filled and they make zero contribution to the total angular momentum. Therefore, the Λ-doubling parameters for the X2Π−A2Σ+ interaction becomes

(20)

where χv(R) is the wave function for the vibrational state v. 8627



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Table 3. Spectroscopic Constants for the Lowest Four Electronic States of NH− and the Ground Electronic State of NH species −

state

method

XΠ

a

2

NH

MRCI/CBS PNO-CEPA22 MC(11)-CI24 SCEP-CEPA24 exp17 MRCI/V6Za MRCI/V6Za MC(12)-CI24 MRCI/CBSa MRCI/CBSa MRSD-CI41 exp42

A′2Σ− A2Σ+ a4Σ− X3Σ−

NH

energy (au)

Re (Å)

ωe (cm−1)

ωexe (cm−1)

Be (cm−1)

αe (cm−1)

De (cm−1)

Te (cm−1)

−55.15653798 −55.1370 −55.1442 −55.1532

1.0345 1.039 1.045 1.043 1.039 1.048 1.0305 1.040 1.061 1.0365 1.0398 1.0372

3244 3226 3155 3173 3020.36b 3095 3238 3157 2817 3288 3198.6 3266

87.62 85 88 89

16.55 16.61 16.41 16.48 16.607 16.28 16.82 16.57 15.78 16.70 16.7 16.67

0.734 0.691 0.876 0.731 0.712 0.717 0.745 0.739 0.714 0.739 0.641 0.646

0.001696

0.0 0.0 0.0 0.0 0.0 18104 31506 29153 15380 0.0 0.0 0.0

−55.07404708 −55.01297956 −55.0117 −55.08645846 −55.15552971 55.136 030

82.83 89.62 85 86.89 80.06 82.71 78.5

0.001756 0.001814 0.00211 0.00161

Calculated by fitting vibrational and rotational states to the equation Ev,J = (v + 1/2)ωe − (v + 1/2)2ωexe + BeJ(J + 1) − αe(v + 1/2)J(J + 1) − De[J(J + 1)]2. bΔG1/2 value estimated from the fit of the R branch in Neumark et al.17 as taken from Mänz et al.24 a

a+ = ⟨π +|a l̂ +|σ ⟩ = A vΠ 2

(27)

b = ⟨π +|l+|σ ⟩ =

(28)

2

Using eqs 16, 17, 25, 26, 27, and 28, we can calculate the contribution of the A′2Σ− state to the Λ-doubling parameters pv and qv as

(22)

2

(23) 2 −

The contribution of the A′ Σ state cannot be obtained directly from the pure precession approximation because this configuration cannot be written as a σ1 or a π1 configuration. The relationship for such a complex configuration can be derived in terms of a generalized pure precession. We can express the A′2Σ−(π2σ1) state in terms of Slater determinants leaving all fully filled orbitals as |2 Σ−⟩ =

1 [2|π +απ −ασ *β| − |π +απ −βσ *α| 6 − |π +βπ −ασ *α|]

i

⟨2 Π1/2 , v |B ∑ li+|2 Σ−, v′⟩ = + i

1 a+⟨v|v′⟩ 6

3 bBvv ′ 2

(24)

(25)

(26)

where B = ℏ /2μR . Now, using the pure precession approximation, we can write 2

(30)

v

pΠv (2Σ+)

pΠv (2Σ−)

pΠv

qΠv (2Σ+)

qΠv (2Σ−)

qΠv

0 1 0 1 0 1

0.1112 0.1078 0.09219 0.08413 0.089 0.081

−0.1057 −0.1042 Al Zaʼal et al.19

0.0055 0.0036

−0.0333 −0.0304 −0.0212 −0.01816 −0.0213 −0.0182

0.0949 0.0881

0.0616 0.0577

Neumark et al.17

values. As expected, the calculated values of pv and qv with contributions from the A2Σ+ state are in agreement with the experimental values obtained by fitting a Hamiltonian that includes only the 2Σ+ state. The contribution from the 2Σ− state is of opposite sign to that from 2Σ+. Therefore, the overall values of pv and qv are much lower than the experimentally reported values and are positive. Having obtained the Λ-doubling parameters, we went on to calculate the elements of the interaction matrix (9). The matrix is then diagonalized to obtain the fine splitting including the Λ-doubling for each Ω component of the ground state (X2Π) of NH−. The schematic diagram in Figure 5 shows the energy values obtained in this calculation for the vibrational quantum numbers 0 and 1. It has been reported by Neumark et al.17 and Al-Zaʼal et al.19 that the fine structure of NH− is intermediate between Hundʼs coupling case a and case b. In Hundʼs case a, the electronic angular momentum Ω is well-defined and the

Therefore, the required matrix elements from eqs 16 and 17 are written as ⟨2 Π1/2 , v |∑ aî li+si−|2 Σ−, v′⟩ = +

qv(2 Σ−) = −6Bv 2 /ΔE ΠΣ

Table 4. Λ-Doubling Parameters for the Ground Electronic State of NH−

2

qv( Σ+) = 4Bv 2 /ΔE ΠΣ

(29)

The details of the derivation of these equations can be found in ref 29. The values obtained for the Λ-doubling parameters with individual contributions from the A′2Σ− and A2Σ+ states are given in Table 4 along with the reported experimental

Figure 4. Spin−orbit coupling constant (Ae) as a function of the internuclear distance (R) for NH− in its (X2Π) state at the MRCI/ AV6Z level of theory.

pv ( Σ+) = 4A v Bv /ΔE ΠΣ

pv (2 Σ−) = −2A v Bv /ΔE ΠΣ

2

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Because for large values of J, Hundʼs case b is applicable, we have J = N + 0.5 for the F1 manifold and J = N − 0.5 for the F2 manifold. The Λ-doubling levels with parity (−1)J−1/2 are labeled as e and those with parity −(−1)J−1/2 are labeled as f, according to the convention proposed by Brown et al.46 The selection rule for parity changes for the P, Q and R branches is as follows: e↔f

for

e↔e

or

ΔJ = 0(Q ) f↔f

for

(31)

ΔJ = ±1 (P and R )

(32)

Transition energy values obtained for all the three cases referred to above are reported in Table T1 in the Supporting Information. The P1, Q1, and R1 transitions correspond to Ω = 3/2 and the P2, Q2, and R2 transitions correspond to Ω = 1/2. A comparison of the computed transition energy values with the experimental values in Table T1 (Supporting Information) provides some insight into the spectrum and the influence of the 2Σ± excited states. The largest difference between the theoretical and experimental values is observed for the larger J values in the R branch of the spectrum, whereas the best agreement between theory and experiment is found in the Q branch of both the spin−orbit (2Π3/2 and 2Π1/2) states as illustrated in Figure 6a,b.

Figure 5. Rotational energy levels of the ground (v = 0) and the first excited (v = 1) vibrational level of the X2Π state of NH− along with the splitting arising from the perturbation due to the A2Σ+ and A′2Σ− states. The splitting due to Λ-doubling is exaggerated for the sake of clarity. Otherwise the figure is drawn to scale. (Left panel of the figure is similar to Figure 2 of Al-Zaʼal et al.19).

total angular momentum J is a resultant of Ω and the nuclear rotation angular momentum N. In Hundʼs case b, the spin angular momentum S decouples with the z-component (along the internuclear axis) of the orbital angular momentum Λ and hence Ω is not defined. In this case, Λ and N together form a resultant K, which then couples with S to form a resultant J.29,32,45 For low J values, NH− follows Hundʼs case a coupling scheme forming spin−orbit states 2Π3/2 and 2Π1/2 denoted by F1 and F2 manifold, respectively. For large values of J, the coupling becomes Hundʼs case b and S decouples with the z-component of Λ along the internuclear axis. This feature was evident in the experimental IR spectrum by the reversal of the order of parity in F2 manifold between J = 1.5 and 2.5. In our calculations we have observed the same behavior with the inclusion of the A2Σ+ state. When the A′2Σ− state is included in the calculation of Λ-doubling, no such reversal of parity in F2(2Π1/2) manifold is observed. The rotational energy levels and their splitting due to Λ-doubling for the first four J values are shown in Figure 5. The Λ-doublets are shown enlarged for the sake of clarity. The left and right panels in Figure 5 show the splitting due to perturbation by both the A2Σ+ and A′2Σ− states and an inversion at J = 2.5 can be seen in the left panel. The reason for such a behavior could be the following. The low energy difference between the interacting vibrational levels of A′2Σ− and X2Π states increases the value of pv and qv. These values are opposite in sign to the values that we get with the inclusion of only the 2Σ+ state. The final value obtained by summing over both contributions is positive for pv and qv. This increases the extent of Λ-doubling in the X2Π state and it also increases the spin decoupling with the orbital angular momentum. As a result, we obtain the energy levels according to Hundʼs case b only rather than switching from case a to case b. We have calculated the energy gap for the v = 0 → 1 transition in the X2Π state under three different conditions: That is, the inclusion of (i) the A′2Σ− state only, (ii) the A2Σ+ state only, and (iii) both states. Because the spin−orbit coupling constant is negative, we get inverted energy levels for the X2Π spin−orbit states. That is, the F2 manifold has a higher energy than the F1 manifold.

Figure 6. Comparison of the computed transition energy values for (a) the R1 branch and (b) the Q1 branch of the rotational fine structure of NH− with the experimental results of Al-Zaʼal et al.19

It is found that the inclusion of the 2Σ+ state results in the correct order of e ↔ e and f ↔ f transitions for a given J 8629



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observed experimentally. The order is reversed for the same J value when only the 2Σ− state is considered or when both 2Σ+ and 2Σ− states are considered. The predicted Q1 and Q2 branches are in good agreement with the experimental values for J > 5.5 for the case where both 2Σ+ and 2Σ− states are included in the interaction. It can be said that it is the overall effect of these two states that is observed experimentally and the two states have opposite effects on the Λ-doubling of the ground state of NH−. Electron Affinity of NH. It has been reported17,19 that the P and Q branch transitions from the v = 0 to v = 1 state of the Ω = 3/2, J = 1.5, f level was observed in the experiment, but the transition to the e state was not observed. Therefore, it was assumed that the lowest level of NH was between the e and f states of the anion. We have calculated the frequency value for the Q1, Ω = 3/2, J = 1.5, e → f transition under the influence of the (i) 2Σ+ state only, (ii) 2Σ− state only, and (iii) 2Σ+ + 2Σ− states and found the electron affinity values 0.375695, 0.375723, and 0.375704 eV, respectively, to be in good agreement with the experimental results of Neumark et al.17 and Al-Zaʼal et al.19 A comparison of the computed electron affinity value with the theoretical and experimental results is given in Table 5.

where the subscript X indicates the cardinal numbers (D = 2; T = 3; Q = 4) in AVXZ basis. The superscripts CAS and dc stand for complete-active-space and dynamical correlation energy, respectively. The CAS energy is extrapolated using the following three parameter formula47 with cardinal numbers X = Q, 5, and 6: CAS E XCAS = E∞ + Ae(−bX )

where the parameters ECAS ∞ , A, and b are obtained through a least-squares fit. The dynamical correlation energy is extrapolated by the formula dc E Xdc = E∞ + A3Y

0.375695 0.375723 0.375704 −0.25 0.01 0.22 0.37 0.376 0.38 ± 0.03 0.381 ± 0.014 0.370 ± 0.004 0.374363 ± 0.000005

method with 2Σ− perturbation with 2Σ+ perturbation with 2Σ− + 2Σ+ perturbation PNO−CI (theor) CEPA (theor) united atom (theor) separated atom (theor) HFDFT/WMR (theor) exp exp exp exp

(35)

The variable Y is defined as Y = (X + α)−3 [1 + τ53(X + α)−2 ]

(36)

τ53 = A5 /A3

(37)

A5 = A5(0) + cA3n

(38)

where A5(0) is the basis-set error in energy for a pure pair− triplet interaction, and c and n are least-squares parameters. The values of these parameters have been taken from the literature38 as α = −3/8, A5(0) = 0.0037685459 Eh, c = −1.17847713Eh1−n, and n = 1.25.

Table 5. Electron Affinity of NH electron affinity (eV)

(34)

■

reference present work present work present work

ASSOCIATED CONTENT

S Supporting Information *

Figures of values of (a) ⟨r2⟩1/2, (b) ⟨x2⟩1/2, (c) ⟨y2⟩1/2, and (d) ⟨z2⟩1/2 obtained at the MRCI level of theory using AVXZ and VXZ (X = D, T, Q, 5, 6) basis sets for the A2Σ+ and a4Σ− states of NH−. Table of rovibrational transitions for the ground electronic state of NH. This material is available free of charge via the Internet at http://pubs.acs.org.

Rosmus and Meyer22 Rosmus and Meyer22 Cade et al.21 Cade et al.21 Gutsev and Bartlett25 Cellota et al.15 Engelking and Lineberger16 Neumark et al.17 Al-Zaʼal et al.19

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

■

ACKNOWLEDGMENTS We are grateful to Professor Antonio Varandas (Universidade de Coimbra) for initiating us into the investigation of diatomic anions. N.S. is an honorary professor at the Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore. N.S. thanks the Department of Science and Technology, New Delhi, for a J. C. Bose Fellowship. S.S. thanks the Council of Scientific and Industrial Research, New Delhi, for a Senior Research Fellowship.

CONCLUSIONS Accurate potential energy curves have been obtained for the ground (X2Π) and excited states (A′2Σ−, A2Σ+, and a4Σ−) of NH− and the ground (X3Σ−) state of NH. The second moment of position has been used as a tool to identify the correct basis set for the ab initio calculation for the resonance states of NH− buried in the continuum of the NH + e system. The rotational fine splitting for the ground state of NH− has been calculated and the Λ-doubling for the first and second vibrational states has been investigated. It is found that the transition from Hundʼs case a to Hundʼs case b arises due to the A2Σ+ state only and this effect is observed in Λ-doubling. The electron affinity of NH has also been computed and found to be in accord with the experimental result.

■

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APPENDIX: CBS EXTRAPOLATION Electronic energy (EX) obtained by the MRCI method can be written as a sum of two terms: E X = E XCAS + E Xdc

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