- Email: [email protected]
Ab initio MRD-CI potential surfaces for the low-lying states of the NH+2 molecular ion
Chemical Physics 42 (1979) 167-176 Q North-Holland Publishing Company
AB INITIO MRD-CI POTENTIAL SURFACES FOR THE LOW-LYING STATES OF THE NH; MOLECULAR ION
Sigrid D. PEYERIMHOFF Lehrstuhl D-5300
fir
Theoretische
Chemie, Uniaersitiit
Bonn,
Bonn, Germany
and Robert J. BUENKER Lehrsruhlflr D-5600
Theoretische
Chetnie. Gesomthochschule
Wuppertal-Elberfild,
Wuppertal,
Germany
Received 21 February 1979
A series ofab initio SCF and MRD-CI calculations are presented for the lowest fourteen states of the NH: ion. Angular and NH symmetric stretch potential curves are obtained and distortions involving asymmerric conformations are also considered. The O-0 energy separating the ground (3B,) and lowest excited (‘A,) state is found to be 29.9 kcal/mole in the calculations; oscillator strengths for transitions connecting each of the latter with higher states of the same multiplicity are also cakulated. The energies of the various fragments of NH: have also been obtained in an equivalent treatment for a variety of states and it is found that the ‘B, state has a dissociation energy of 6.45 eV (0.) while that for the 3AZ,which is thought to be important for NH: formation, is calculated to be 2.79 eV.
1. Introduction
Another area of interest is the electronic spectrufn of NH; since it is assumed [7] to occur in the upper atmosphere and/or interstellar space. Hence the major emphasis in the present work is placed on describing a large number of electronic states in nuclear geometries similar to that of NH; in its equilibrium conformation in order to predict the absorption spectrum of this molecular ion, i.e. the transition energies, the intensities and the vibrational structure of the corresponding bands, and thereby furnish guidelines for the identification of this molecular species from the appearance of its spectrum. The study will be undertaken by means of large-scale configuration interaction treatments in a similar manner as has been employed in an investigation of the isoelectronic CH, molecule [S], with which comparison will be made whenever it appears to be appropriate. Only the H-N-H conformations will be studied in the present work, since the N-H-H conformer generally lies higher in energy [4];
The possible formation of the NH: ion from its constituents Nf plus H, and N plus Hi has been investigated in detail in molecular beam experiments [l, 21, and careful theoretical studies of the N + + H, ion-molecule reactions by ab initio calculations have been reported at various levels of sophistication in both the A0 basis set constitution and configuration interaction treatment employed [3-51. The theoretical work finds the ion to be most stable in its symmetric HNH form in the ‘B, electronic state and predicts an intersystem crossing of ‘B, and ‘A, potential surfaces (in a C, perturbation) to play the dominant role in formation of NH: from the NC(3P.J and H,(‘C:) fragments. Further calculations have also considered the formation of NH; from the N atom and the Hi ion as weil as in the outgoing NH(4X-) + H channel via a quintuplet state [6]. 167
168
S.D. Peyerimhofi R.J. Buenker/Potential sur/nces for the lorv-lying stores oJNHl
the various separated species N’, N, H2 and H,C will also be calculated as reference points to be combined with earlier theoretical results for the lowenergy dissociation surfaces of the NH; system
Table 1 Electronic states of NH: for which SCF calculations have
been carried out State No.
2. Technical details of the treatment The A0 basis set employed in the present work consists of the (9s. 5p) set for nitrogen given by Huzindga [9] in the (5s, 3p) contraction recommended by Dunning [lo], in addition to two nitrogen d functions with exponents a,(d,) = 1.8846 and a,(d,) = 0.5582 which have already been employed in an earlier treatment of the Nz molecule [I I]. The hydrogen A0 set consists of a five-term expansion [12] for the 1s function with scaling factor t)’ = 2.0 in a two-term contraction and a one-component representation of a 2p species with exponent a(p,,) = 0.735. Further polarization is taken into account via two bond functions of s and p character each [c&J = 1.8, c&,J =‘0.14, a,(~,) = 1.5, a,(p,) = 0.183 placed in the center of each of the two NH bonds. Thus the entire A0 basis consists of 52 contracted gaussian functions, which are essentially equivalent to those employed in the related treatment of CH, [S]. For a few geometrical conformations additional long-range Rydberg-type functions are also added [n(p) = a(s) = 0.021 in order to ascertain whether such species are important for the higher-energy states under discussion. The configuration interaction treatment is of the general MRD-CI type [13, 141 with configuration selection and energy extrapolation [14]. A core of one orbital corresponding to the Is inner shell is kept doubly occupied in all contigurations and one MO representing the inner-shell complement is disregarded entirely in the treatment, while all other (50) orbitals are allowed variable electron occupation. Self-consistent field solutions are obtained for all the low-energy states given in table 1 which are accessible via the Roothaan method [lo]. These MO’s are generally employed as the basis for the CI calculations, details of which [number of main (or reference) configurations from which single- and double-excitation species are formed, secular equations actually solved (selected configurations)
CorrclationtolinearNH;
Contiguration
WI.
&i, Ibz 3a, Ib, 4a,
.-
2Q 2eU II, 30, -
X’B, $22 I
2
22
1I
1. I
5;
2
z
‘A, 3
2
2
Z
0
‘4
2
2
2
‘A, 4 2 ‘A2 52 ‘B, 6 2
2
0
2
’ “;
2
1
12. 1 1 I 2
2
‘IT,
2
2 I
3
‘%
2
I
3
‘A;72 ‘FL82
I
I
I
I2
z
and configuration spaces for which the CI energy has been extrapolated] are summarized in table 2 for C2, nuclear conformations. All configurations which contribute more than 0.25 “/, (on a c* basis) to the final CI wavefunction expansion are thereby included in the reference set. Finally an estimate of the full CI energy (within the A0 basis set given) is
Table 2 Detaits of the CI treatment” undertaken for the various states of NH,+ State
No.
MO Contiguration basis generation/ selection
Secular equations generated/solved
‘B,
I
3B,
‘Bt ‘A, 2’A, 4lA, 3A, 3B, ‘A, ‘Bz 5A2 ‘AI 23B, 3lA, 2’B,
2 3 4 9 5 6 7 8 10 II 13 14 IS
‘B,
IMlR IMIR
8426/l 933-2624 5454/1528-2277
IA,
8M3R
20034/1866-3177
“A, 3B, ‘A, ‘B2 5A, ‘4
lMlR lM1R 1MlR lM1R 2MlR 3MlR
IA, 3A, 5A,
4M2R 7MlR
8525/1723-2555 757911692-2326 5496/1407-2198 5041/1386-2036 15702/1433-1982 19610/1985-2501 52475/2017-21 I1 14179/2576-3299 31972/1739-2005
7MlR
continued on following page
S.D. Peyeriothofi R.J. Buenker/PotentiaI srr$uces for the low-lying stares of NH; Table 2 (continued) State
No.
Conftgurations (occupation) ?a, IbZ 3a, lb, 4a, 2b, 3b, 2b, Sa,
3,4,9
2 2 2 2 2 2 2 I 2 I 2 2 22-l-l 22-l--1 2 -21--l
2 2 2 2
‘AZ 10
2 1 1 1211---l
1
1
3A,
2
2 2
1 -
1 1
1 -
1
1
1 1 1
1 1 I
1
1
-
1
2 2
1 1 1
‘A,
11
1
=B,
13.15 (1.2)
2 2 2 1 2 2
3lA,
14 (3)
2 2 1 -
2 2 2
1
2 2 2 2
2 2 I
1 1
1
2 2 2 2
2 1
1
1
I 1
1
1
” The term nMmR denotes that n main or reference conligurations are employed for generation of the configuration space (to which the MRD-CI energy is extrapolated) while configuration selection is undertaken for m roots.
169
study obtains for its optimal geometrical parameters NH = 1.940, and L HNH = 150” (CI value’). The calculated ground-state SCF energy (at NH = 1.940775a0, L HNH = 1SS’) is -55.22321 hartree, a value which is very close to that calculated by Bender et al. [3] (-55.22965 hartree at L HNH = 143.3” and NH = 1.9238a,) employing a near Hartree-Fock basis of N (13sSp3d/9s6p3d) and H (6s2p/4s2p) functions. The corresponding CI energy in the present work is - 55.3755 hartree at the MRD-CI level and -55.382 hartree for the full-C1 estimate. The molecule must be considered as quasi-linear in its ‘B, ground state since the barrier (difference of total electronic energies at L HNH = 155” and 180’) to linearity is found to be only 330 cm-’ at the full-C1 limit (60 cm-’ more at the MRD-CI level), while the zero vibrational level for the bending frequency is calculated to be of practically the same magnitude (318 cm-‘, K = 0). The SCF treatment yields a barrier approximately three times as large. Comparison with the calculated energy for the neutral NH, species in its equilibrium (‘B, state, NH = 1.935106a,, L HNH = 103.40”$ yields an energy separation (neglecting vibrational effects) between the X3B, ion and the neutral X’B, species of T, = 10.9 eV, while the calculated urrrical (electronic) energy difference is found to be 11.6 eV (only 10.4 eV at the SCF level of treatment), a value which compares quite satisfactorily with the measured IP of 11.4 eV determined from massspectrometric data [17].
undertaken according to the generalization AE = (1 - Cc&%,r
- L,,,_cr),
of a formula given by Davidson [16] derived for the contribution of higher than double-excitation configurations. In virtually all cases the sum of the squares of coefficients for the reference species Zci is found to lie between 93 % and 96 %_
3. Potential energy surfaces 3.1. The NH: ground stute The ground state of NH: is 3B, with the electronic conliguration la:2aflb$3a, lb, ; the present
3.2. Atzgular potential curves The various angular potential curves obtained from the CI calculations at a fixed NH separation of NH = 1.940775a, for the lowest-energy states of NHf are contained in fig. 1; the calculations are generally carried out for internuclear angles of 180, 165,155,140,125,115,105,90,75, and 60 degrees, and these are complemented in some cases at additional angles of 45” and 30”. The various optimal geometrical parameters obtained from ’ The corresponding SCF calculations yield a somewhat smaller internuclear angle of 145”. ’ E(SCF) = -55.57995 hartree, E(MRD-CI) = -55.77318 hartree and E(full-CI est.) = -55.7839 hartree.
170
S.D. PeJleritnho/j. R.J. BuenkerlPotential surfaces/or the
loiv-lying srares ofNH;
Table 3 Calculated internuclear angles and corresponding energies of NH; obtained from polynomial fits to the curves in lig. 1 (The NH distance is 1.940775no throughout.) State
Energy (eV)
‘B, ‘A,
149.6 107.6 155.2 180 60 63 89 98
I%
-N?o”‘.H; -N’klpi2
2 ‘A, ‘A, ‘A, ‘Bz ‘B, -‘Ai 2’B,
0.0
1.29 2.03 3.45 4.12 5.64 7.42 9.24 Il.09
‘H”
-N&&n; -ril+gl.HI
4 ‘A,
180” Z.50
14.2 zs.5
a’ The calculations also show a second minimum around 55”.
Fig. 1. Angular potential curves for a series of NH: states obtained at the (estimated) full-Cl level of treatment (RNH = 1.940775~~);the energies of various dissociation limits (for optimum Hz and Hi bond lengths, respectively) are also indicated. Energy values are given in units of
hartree, HNH angles in degrees. polynomial fits to the data points of fig. 1 are contained in table 3, together with pertinent energy values. It is obvious that the three lowest states, 3B1, ‘A, and ‘B,, are quite similar in character to the corresponding well-known states in the isoelectronic CH, molecule. The equilibrium angles in NH: are all about 5” larger than in CH, [8] (‘A,, 103”; 3B,, 135”; IB,, 140”) and the angular potential curves are more shallow in NH:, although their relative forms are in close analogy to those found for CH,, including the characteristic that the ‘B, internuclear angle is approximateiy 5” larger than that of the corresponding triplet state. This general similarity between NH: and CHI states has already been predicted on the basis of simple MO theory in a
refined Mulliken-Walsh model [18,12,19]. The splitting between ‘A, and 3B, is calculated to be AE, = 29.9 kcal/mole; this value is considerably larger than for the equivalent energy separation in CH,, which ranges in the most recent literature [S, 20-221 from 8.0 to 11 kcal/mole (although photodetachment measurements [23] yield a value of 19.5 + 0.7 kcal/moleT). A two-configuration description of the *A, state relative to the singleconfiguration (SCF) treatment of the 3B, species yields a value for the singlet-triplet splitting of 29.1 kcal[3] in a near Hartree-Fock A0 basis, demonstrating that the dominant parts of the CI expansions of the two states under discussion can be accurately represented by two and one terms, respectively. Other theoretical ab initio predictions [24-261 range from 36 to 45 kcal, while a semiempirical MIND0/3 treatment [27] yields 31 kcal for this quantity. As in CH, the SCF singleconfiguration treatment overestimates this singlettriplet energy splitting and results in the present A0 1
Since all recent theoretical results for this quantity as well as those obtained in thermochemical studies lie in the 8-l 1 kcal area, the interpretation of the photodetechment experiments [23] is presently under extensive study in various laboratories.
S.D. Pqerintltoff, R.J. Buenker/Potential sudaces for the low-lyingSates of NH:
basis in a value of 44.3 kcal, in close proximity to the 44.7 kcal reported in ref. [3] using an extended A0 basis. The second ‘A, species is found to be linear, and this geometry is also expected for the corresponding state of CH,, which has received relatively little attention so far. The two states correlating with ‘fls and ‘H,, respectively, prefer bent geometries since they depopulate the cU-b2 orbital with its strong preference for the linear arrangement of nuclei [ 193. This situation is especially apparent for the states of A, symmetry with a la:2a:lb,3aflb, electronic conliguration since they populate the 3a, MO (which distinctly prefers the bent conformation [l9]) twice in contrast to the Bz states with laf2a:l b23ar 1b: occupation which have the essentially non-bonding MO lb, doubly occupied. The actual energy dillerence between 3Az at its optimal angle (table 3) and 3Bi at an NH distance of 1.940775a, is 4.12 eV (97 kcal) but, as will be seen in the next section, the former is considerably more stable at larger NH bond distances. This effect is even more enhanced in the 4&A, state, in which the lb, MO is not occupied at all; its potential energy curve shows a deep minimum at very small internuclear angle, and leads to avoided crossings with other states of ‘Al symmetry in energy regions for which this state is quite stable (fig. 1). The higher states 3~‘H,(1tr~20~ 1~~~7r~3tr.Jand 5Z-(lo,22u,Z lo,x,23a,J all populate the 3oe4a, MO (taile 1). Since the 4a, species would be a Rydbergtype orbital in neutral AH, systems, supplementary SCF calculations are carried out (at angle 140”) in a basis containing additional long-range 3s- and 3p-type [U(S) = 0.02, EL(P)= Cl.021functions in order to check whether they make any contribution to the 4ar MO in NH:. Not surprisingly the total energies of the first nine SCF states (table I) are lowered only a small amount (O.tlOOO6 hartree, less than 0.002 eV) as a result of this A0 basis extension, but !arger effects are noted for the higher states which occupy 4a,, namely 0.066 eV, 0.072 eV and 0.154 eV, respectively, for the 5A, (IO), 3B2 (12) and ‘A, (11) states of table 1. Nevertheless, such changes seem small enough to justify omission of Rydberg-type functions for the present investigation. The preferred linear geometry in the 3*rHUstates can also be rationalized in terms of MO theory: the
171
effect of doubly occupying the 1b2 orbital (which is considerably more stable for L HNH = 180” [ 191) cannot be offset with only single occupation of the 3a, MO with its bent tendency, especially since the orbital energy of the 4a, species does not vary greatly with change in internuclear angle. In a similar manner the 5A, state shows an energy minimum for the linear nuclear framework, but with a much smaller bending force constant than for the 3-1HUstates; the latter distinction is clearly traceable to the distinction in the lb, occupation numbers of the 5A, relative to these lower-lying states. At smaller internuclear angles the combined tendency of the 3a, and lb, species, which prefer bent NH, geometries, apparently overcomes the influence of the 1b, MO, producing a second minimum at a very small internuclear angle (fig. 1). 3.3. Sytnmetric NH stretch potential curves The potential energy curves for symmetric stretch are contained in figs. 2a, and 2b; they are calculated at the SCF level of treatment for only two internuclear angles: 140” (close to the X’B, minimum) and 105” (which is near the minimum of the lowest NH: singlet state). Similar to the situation for the angular potential energy curves it is seen that the electronic states can be separated into three major groups according to their occupation of the (3a,, lb,), lb, and 4a, MO’s, respectively.’ From earlier calculations [ 191 it is known that the lb, and 3a, MO’s in AH, systems show approximately the same NH bonding behavior and hence the states 3B,, ‘B,, ‘A, and 2’A, show very little distinction in their equilibrium bond lengths (figs. 2a, b and also table 4, which shows the results of a polynomial fit to the data points of lig. 2). The lb, MO is found to be more bonding [19] than 3a, or 1b, and hence depopulating this MO, as is done in the A2 and B, states, leads to the increase in bond lengths obvious from fg. 2. Finally, occupation of the strongly NH antibonding 4a, orbital produces a very large NH equilibrium separation or even a completely repulsive potential curve; such a fragmentation is already obtained at the SCF levei of treatment for the ‘A2 state, for example, which dissociates into N(%,) -t Hz(‘Z:).
S.D. Peperimhc& R.J. BuenkerjPotential surfaces/or the low-lying states of NH;
172
mar1rL
-5L.70
-5L.80
-54.90
-55.m
-55.10
-55.10
-55.x
- 55.20
i.eb
* 2.04
2.21
2.U
NH
laoI
Fig. 2. Symmetric NH stretch potential curves obtained at the SCF level of treatment for fixed HNH internuclear angles of (a) 105” and(b) 140”; the energy of the most stable N+ + H2 dissociation product is aIso indicated.
3.4. Potentiul curves@
antisyrmetric
stretch
Potential curves for asymmetric stretch are only calculated in the immediate neighborhood of the potential minima obtained thus far, with the goal of ascertaining whether the various states prefer (locally) symmetric nuclear arrangements; in addition, approximate values for the antisymmetric vibrational frequencies in these states are also obtained. The results of these (SCF) calculations are collected in table 5 for a series of electronic states. It is found that the low-energy species 3B,, ‘B,, ‘A,, 2’A, and 3B, are all stabie with respect to such a C, distortion around their respective potential minima. Such is not the case, however, for
the ‘A, state which populates the 4a, MO; the results of table 5 show furthermore that this distortion is somewhat more preferred at small internuclear angles than for nearly linear HNH arrangements. Taken together with the results of the previous section it seems very likely that this state (along with the closely related species 3 ‘A, and 23*1B,) is not a stable entity. The 3A, and ‘A, states show a definite increase in energy for small antisymmetric distortion (AR = 0.1 bohi), but exhibit more dissociative characteristics after passing over this barrier; this behavior is also consistent with results of earlier investigations of the ‘AZ potential surface [3,4].
S.D. Peyerimhofi
R.J. BuenkerjPotential
Table 4 Calculated equilibrium separations of NH: (symmetric arrangement) and corresponding energy differences from the conformation employing R,, = 1.940775a,; the values are obtained from a polynomial tit to the SCF data of tig 2
surfacesfor
2
R,
AE(R-
AE(R - R,)
R,) R,
% ‘A,
1.940
-
1.925
-
1.957
-
1.934
-
‘BI
1.962
-
1.933
-
‘A, 3Az
1.989
0.02 eV
1.943
-
2.210 2.247 2.193 2.205 _b) -
0.56eV 0.71 0.51 0.55 -
IA, 3B* ‘Bz 3A, 5Az 4’A,
a) See fig. 2a.
0.47 0.55 0.39 0.38 >2.80=’ > 4.9 >2.5
-
” See fig. 2b.
3.X Separutel1 products
R(a,)
0
1.940775 1.940775 1.940775 1.940775 1.940775 1.940775
155” I55O 155” 155” 155” 155” 105” 105” lOS0 105” 105” 105” 90”
AR=O.lu,
AR=O.Za,
0 = 140
0 = 105”
2.18 2.20 2.156 2.150 > 2.54 > 2.54 > 2.54
173
Table 5‘ SCF energy changes (in eV) for antisymmetric distortion of the NH,+ framework relative to a given conformation (R, f-J State
State
the low-lying states of NH;
[N
+ H,]
+
The energies of the separated products N+ plus Hz as well as N plus Hl have also been calculated in the present work as reference points for the states of the combined NHf system. In order to provide for a balanced treatment of the combined molecule and the separated fragments, the calculations for the former species are carried out at an internuclear distance of lOa, between the nitrogen nucleus and the H, framework (with the appropriate H-H separations, 2.003a, for Hf and 1_40a, for HJ in an equivalent manner as for NH: near its equilibrium nuclear arrangement. The N(4S,) + Hf(‘Zl) complex can be obtained in Czv symmetry either from a calculation of NH: for a ‘A, or a 3Az multiplet, and the results in table 6 show that there is good numerical agreement in the two technically different calculations; details of the CI treatments are contained in table 7. The corresponding singlet state ‘A, corresponds to N*(‘D,) + H:(*Z:) and is calculated to lie 2.63 eV higher; the experimental N(4S,)-N(2D,) excitation energy is 2.38 eV [28], but it has already been found in an earlier CI study on the dissociation of N, in ground and excited states that the present A0 basis yields a 4SU-*D, splitting which is too large by 0.23 eV.
jB, ‘A,
.‘B, 2 ‘A, gB, 3A, 3B, ‘A, ‘B, 2 ‘A,
3Bz JA* 3BZ
1.940775 1.940775
1.940775 1.940775 1.940775 1.940775 2.140775
0.18 0.19 0.19 0.17 0.23 -0.43 0.11 0.12 0.12 0.10 0.10 -0.54 0.04
0.49 0.49 0.49 0.46 0.49 0.47 _ 0.17
The N+ + Hz fragments can also be obtained in various ways in a formal NH: molecular calculation; details are also given in tables 6 and 7. The ‘B,, 3A, and 3B, states all correlate with the lowest N+(3P,) + H&X:) fragment states and table 6 again shows that there is good numerical agreement in the various treatments when various MO basis sets and configuration spaces are employed. The singlet states ‘B,, ‘A?, and ‘B, of equivalent character as the triplet species correlate with the first excited state of the nitrogen ion, N+(’ D,) + H,(‘Zl). The present calculations lind an energy difference of 2.09 eV for this 1D,-3P, splitting; the corresponding experimental result is 1.89 eV [23].’ Comparison with previous work can also be made at this point. The SCF calculations of Bender et al. [3] yield an N+(3P,) + Hz energy (3A2 state) of - 55.02 123 hartree in a near Hartree-Fock basis and a value of - 55.01159 hartree in their DZ + P basis, while the present work finds an SCF energy of -55.01363 hartree for the same fragments. The CI energy obtained in ref. [3] (employing the vector method) in the DZ + P basis is -55.12329 hartree, recovering 0.1117 hartree of the valence-shell correlation, whereas the present CI energy (MRD-CI level) is -55.13898 hartree, corresponding to a correlation energy of 0.12535 hartree; the total valence-shell correlation energy of the present basis is estimated to be 0.13 18 hartree (full-C1 estimzte for
S.D. Pegeridtofi
174
R.J. BuenkerjPotential surfices for the low-lying states of NH2+
Table 6 Total energies (hartree) obtained for the various [N-H,]’ species at a distance between nitrogen and the Hz framework of lo&,; the correlation with the separated fragments is also indicated State
Calc. No.“’
Fragments
SCF
- 54.99426
MRD-CI
Full CI estimate
- 55.09994 - 55.09975 - 55.0025
-55.1036 -55.1035 - 55.0070
HH = Z.Oa,
5A, 3Az
1
N(4S,) + H:(‘Z;)
‘A2
2 3
N(%,,) •t Hz’@;) N(*D,) + Hf(‘Z,C)
HH = 1.4a. 3&
4
‘& 3A LAS
2 I8
N+(‘PJ N’(3P,) N+(‘D,) N+(‘D,) N+(3P,)
+ H&Y;) -I- H,(‘Z;) + HJ’z;) +c HJ’Z;) H&T)
_ - 54.933101 T 55.013629
-55.13839 -55.14005 -55.06130 - 55.13898 -55.06179
-55.1454 -55.1449 - 55.0685 --55.0683 55.1454
382
9 10 11 12
N+(‘P,) N’(3P,) Ni(‘D,) N’(‘D,)
+ + + +
-54.01359 -54.01359
-55.13900 -55.14005 -55.06157 - 55.06348
-55.1454 -55.1449 -55.068? - 55.0685
IB,
-
H,(‘Z;) H1(‘E;) H,(‘E;) H,(‘E;)
-
‘) Calculation.No. refers to the treatment explained in table 7.
Hz + N’(3P,) is -55.1454 hartree; table 6). The calculations of_Gittins et al. [4] obtain -55.06728 hartree [30] in an essentially DZ basis and the CI treatment of Hirst [SJ yields -55.12846 hartree in a DZ + P A0 basis. Consideration of the results for the X3B, groundstate and the lowest-lying set of fragments Table 7 Details of the CI treatment carried out for the various [N-H?]+ species at a distance between nitrogen and the H, framework (C,, approach) of lOao representing effectively separated fragments (notation as in table 2) State
Calc. No.
MO basis
Configuration generation/ selection
Secular equations solved
5A2 3A, ‘A2
1 2 3
5A, SCF ‘A, SCF ‘A,SCF
2MlR 1MlR 1MlR
15702/ 919 18405/1517 10915/1206
4 5 6 7 8 9 10 II 12
5AzSCF ‘A, SCF ‘A2 SCF ‘A2SCF ‘AZ SCF ‘Bz SCF 3B2 SCF 3B, SCF 3B, SCF
1MlR 2MlR 1MlR IMIR 1MlR 1MlR 2MlR IMIR 2MlR
8526/ 834 15711/ 852 5544f 699 8525f 675 5496J 557 9135/ 638 15711/ 852 6GO6/ 513 10159/ 712
N’CP.J
+ Hr(‘xl) ieads to a dissociation energy of eV; previous theoretical findings for this quantity range from 5.66 eV [30] in a DZ and 5.98 eV in a DZ + P treatment [S]. The potential well for the 3A, state (relative to NC + Hz) is found to be approximately 2.80 eV (65 kcal)‘, which is also slightly deeper than in several previous investigations (1.76 eV [4], 2.63 eV [s] and 60 kcal [3])_ A collection of the most important data pertinent to the correlation between NHf and its [N + Hz]+ fragments is given in table 8 and it is clear from these results as well as those of fig 1 that the correlation diagram given by Fair and Mahan [I], v&h has been widely employed for the interpretation of the mechanism of NH: formation, is in D, = 6.45
need of considerable
‘4 ‘B, 3Az ‘A, =Bz ‘B1
revision
[or quantitative
use.
4. Intensities in the NH: spectrum In order to obtain a more complete picture of the possible absorption (or emission) spectrum of the ’ In this case the energy at equilibrium is determined as the value for the angular minimum obtained from CI calculations at R = 1.94u,, combined wifh the difference in SCF energies for this NH distance and the optimal separation of 2.19ao.
S.D. Peyerinrhojf, R.3. Buenker/Potential sur/acesfor the low-lpirrg states o/NHf
Table 8 Energies (in eV) of the various (bound) NH, states relative to the separated products. The values, unless directly calculated, are estimated from the calculated data in tables 3 and 4, and the corresponding nuclear conformations are given in parentheses State/Configuration N+(3P,) + H#&+) N(%,) -t Hf(“X;) N+(‘DJ + H&X;) N(‘DJ + I&+(*X;,
% ‘A, ‘B, 2 ‘A, 3Az ‘AI 3B2 rBz
A& 0.0 1.14 2.09
3.76 (3.46,see text) -6.45 (IW, 1.94) -5.16 (108”, 1.94) -4.42 (IS, 1.94) -3.02 (ISO”,1.99) -2.79 (60”,2.18) - 1.36(63”,2.20) 0.58 (89q 2.16) 2.41 (98”.2.!6)
NH: ion the intensities for transition between the states in fig. 1 are also of importance. Hence the electronic transition moments andfvalues between the upper triplet states and X3B, are determined at L HNH = 125”, which represents an average equilibrium angle for the states in question+. The transition moments connecting the upper singlet species with the lowest singlet NH; state are determined at L HNH = 105”, since this angle is close to the equilibrium value of the lowest ‘A, state. In all cases the MO’s of the upper state (see table 2) are generally employed in obtaining the ground state (3B, or ‘A,) CI expansion so as to avoid evaluating the respective matrix elements in a non-orthogonal MO basis. For comparison with the triplet results the corresponding singlet transitions have also been calculated at 125”. The resulting intensity data are summarized in table 9, together with the corresponding vertical electronic energies at the given angle (hence this energy is only approximately equal to the vertical excitation energy from the lower-state equilibrium conformation). It is seen that the 3A,-3B, transition is relatively weak and that thefvalue obtained is quite consis-
’ Note that at 180” the interesting %Is-%;
transition is forbidden under the dipole selection rules, and also that the calculated equilibrium angle for ‘B, is 150”.
175
tent in three different MO basis sets. Furthermore, it is clear that the different forms of the 3A, and ‘B, potential curves (fig. 1) will effect a long progression in the bending vibration for this electronic system. As a result the ‘B,A~A, transition is not expected to occur as a strong feature in the NH: spectrum. The 3A1-3B1 transition moment matrix element is quite large (table 9), but since 3A, exhibits a repulsive potential curve along the N-H stretching coordinate a long progression corresponding to a dissociative state is expected in this instance; this transition should most likely be observed experimentally as an almost structureless underlying feature in the NH; spectrum at relatively high energy. Of the remaining triplet-triplet excitations the 3B,-3B, species is not allowed according to the Table 9 CalculatedJvaIues for transitions between various states of NH: states (R = 1.940775a,) Transition
0 = 1X0 ‘AZ-‘B ,
y1
0.00088 0.00077
0.00086 3BL-3B, dipole forbidden 3A,-3BI ‘AZ-‘B, 2 ‘A,-‘B, 3 ‘A,-‘B, 4 ‘A,-‘B, *B,-* B,
0.014 0.001 I 0.0020 0.013 0.55-5 dipole
‘B,-‘A,
0.00025
A&,,, (cV)
6.62
MO’s employed to calculate /value
3A1 MO’s 3B, MO’s
‘A, MO’s 7.72 12.6 5.77 2.30 11.37 14.55 7.19
‘A, MO’s ‘A2 MO’s ‘A, MO’s ‘A, MO’s ‘A, MO’s
0.84
‘A, MO’s
1.66 4.73 5.66
‘BI MO’s ‘A, MO’s
forbidden 0 = 1,050 ‘B,-‘A, 2 IA,-‘A, ‘AZ-IA,
0.00086 0.001 I dipole forbidden ‘B,-‘A, 0.6-6 3 ‘A,-‘A, 0.12 4 ‘A,-‘A, 0.15-4 2 ‘A,-‘B, 0.0038 ‘B2-2 ‘AI 0.0010 3 ‘A,-2 ‘A, 0.00086
7.95 13.2 13.44 3.08 3.23 8.57
‘B, ‘A, IA, ‘Br ‘Bz “A,
MO’s MO’s MO’s MO’s MO’s MO’s
‘t The value of the oscillator strength is calculated
according tof= (2/3)l(Y,/ R,.,,. /Yu,)l’ BE. The notation 0.n - p stands ior 0.n x 10-p.
176
S.D.
Peyeritnizoff, R.J. Buenker/Potential surfaces for the low-lying states of NHf
dipole-selection rules and the 23B,-X3B, counterpart corresponds to an even higher transition energy than that of 3A,-‘B, ; since the latter upper state is also repulsive in the NH stretching coordinate, similar to the 3A,, its transition moment relative to 3B, has not been calculated. The corresponding transition from ‘A, and 3 ‘A, to the ‘B, state are of similar intensities as those between the corresponding triplets (table 9). The additional 21A,-‘B1 intercombination, which does not possess a triplet counterpart, has anfva!ue of 0.0020, but the transition is also forbidden in the linear nuclear geometry; since the shape of both curves is very similar the Franck-Condon factors are expected to be quite large for intercombination of equivalent vibrational states. _ The transitions to the lowest bent singlet state are also relatively weak for all species which have gerade symmetry in the linear molecule. The uttgerade states such as 3’ A, show a largefvalue for combination with ‘A,, but again as in the triplet case discussed earlier the 3 “Al is very high in energy, lying above the lowest dissociation limits for this system, and shows a predominantly repulsive potential curve towards NH stretch. The oscillator strength of the ‘B,-‘A, transition, on the other hand, is ofsimiiar magnitude as the corresponding ‘BI-‘AI band system in NH2 which is well understood, so that both ‘B,-‘A, and 2lA,-‘A, transitions should be relatively easy to observe in addition to 21A,-‘B1 intercombinations if the NH: system can be found in sufficient concentration to afford measurement of its electronic spectrum.
c31C.F. Bender, J.H. Meadows and H.F. Schaefer III, Faraday Discussions Chem. Sot. 62 (1977) 59. [41 M.A. Gittins, D.M. Hirst and M.F. Guest, Faraday
Discussions Chem. Sot. 62 (1977) 67; M.k Gittins and D.M. Hirst. Chem. Phys. Letters 35 (1975) 534. D.M. Hirst, Mol. Phys. 35 (1978) 1559. ;i; D.M. Hirst, Chem. Phys. Letters 53 (1978) 125. c71G. Herzberg Quart. Rev. 15 (1971) 201; and private communication. PI S. Shih, SD. Peyerimhoff and R.J. Buenker, Chem. Phys. Letters 55 (1978) 206.
PI S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. T-H. Dunning, J. Chem. Phys. 53 (1970) 2823.
;t:; W. Butscher, S. Shih, R.J. Buenker and ?.D. Peyerimhoff, Chem. Phys. Letters 52 (1977) 457.
Cl21S.D. Peyerimhoff, R.J. Buenker and L.C. Allen, J. Chem. Phys. 45 (1966) 734, Appendix.
Cl31R.J. Buenker, S.D. Peyerimhoff and W. Butscher, Mol. Phys. 35 (1978) 771.
Cl43R.J. Buenker and S.D. Peyerimhoff, Theoret. Chim. Acta 35 (1974) 33; 39 (1975) 217.
EN C.C.J. Roothaan, Rev. Mod. Phys. 32 (1960) 179. E.R. Davidson, in: The world of quantum chemistry, II161 eds. R. Daudel and B. Pullmann (Reidel, Dordrecht, 1974) p. 17; S.R. Langhoff and E.R. Davidson, Int. J. Quantum Chem. 8 (1974) 61. Cl71G. Herzberg, Electronic spectra of polyatomic molecules (Van Nostrand, New York, 1966).
Cl81A.D. Waish, J. Chem. Sot. (1953)2260. Cl91R.J. Buenker and SD. Peyerimhoff, Chem. Rev. 74 (1974) 127.
c201R.K. Lengel and R.N. Zare, J. Am. Chem. Sot. 100 (1978) 7495, and references therein. P11 L.B. Harding and W.A. Goddard, Chem. Phys. Letfers 55 (1978) 217.
[22] C.W. Bauschlicher Jr. and I. Shavitt, J. Am. Chem. Sot. 100 (1978) 739.
c231P.F. Zittel, G.B. Ellison, S.V. O’Neil, E. Herbst, W.C. Lineberger and W.P. Reinhardt, J. Am. Chem. Sot. 98 (1976) 3731.
c241S.T. Lee and K. Morokuma. J. Am. Chem. Sot. 93
Acknowledgement
(1971) 6863.
The authors wish to thank the Computing Center of the University of Bonn for services and computer time made availabIe for this study.
cw S.Y. Chu, A.K.Q. Siu and E.F. Hayes, I. Am. Chem. sot. 94 (1972) 2929.
IPI J.F. Harrison and C.W. Eakers. J. Am. Chem. Sot. 95 (1973) 3467.
c271M.J.S. Dewar, R.C. Haddon, W.K. Li, W. Theil and
[=I
References [I] I.A. Fair and B.H. Mahan, J. Chem. Phys. 62 (1975) 515.
[2] D.J. McCIure, C.H. Dou&dss Chem. Phys. 66 (1977) 2079.
and W.R. Gentry, J.
P-K. Weiner, J. Am. Chem. Sot. 97 (1975) 4540. C. Moore, Atomic energy levels, Nat. Bureau of
Standard Circular No. 467 (1949). W. Butscher, S.-K_ Shih, R.J. Buenker and S.D. Peyerimholf, Chem. Phys. Letters 52 (1977) 457. D.M. Hirst, in: General discussion on potential L-301 energy surfaces, Faraday Discussions Chem. Sot. 62 (1977) 138.
m
AB INITIO MRD-CI POTENTIAL SURFACES FOR THE LOW-LYING STATES OF THE NH; MOLECULAR ION
Sigrid D. PEYERIMHOFF Lehrstuhl D-5300
fir
Theoretische
Chemie, Uniaersitiit
Bonn,
Bonn, Germany
and Robert J. BUENKER Lehrsruhlflr D-5600
Theoretische
Chetnie. Gesomthochschule
Wuppertal-Elberfild,
Wuppertal,
Germany
Received 21 February 1979
A series ofab initio SCF and MRD-CI calculations are presented for the lowest fourteen states of the NH: ion. Angular and NH symmetric stretch potential curves are obtained and distortions involving asymmerric conformations are also considered. The O-0 energy separating the ground (3B,) and lowest excited (‘A,) state is found to be 29.9 kcal/mole in the calculations; oscillator strengths for transitions connecting each of the latter with higher states of the same multiplicity are also cakulated. The energies of the various fragments of NH: have also been obtained in an equivalent treatment for a variety of states and it is found that the ‘B, state has a dissociation energy of 6.45 eV (0.) while that for the 3AZ,which is thought to be important for NH: formation, is calculated to be 2.79 eV.
1. Introduction
Another area of interest is the electronic spectrufn of NH; since it is assumed [7] to occur in the upper atmosphere and/or interstellar space. Hence the major emphasis in the present work is placed on describing a large number of electronic states in nuclear geometries similar to that of NH; in its equilibrium conformation in order to predict the absorption spectrum of this molecular ion, i.e. the transition energies, the intensities and the vibrational structure of the corresponding bands, and thereby furnish guidelines for the identification of this molecular species from the appearance of its spectrum. The study will be undertaken by means of large-scale configuration interaction treatments in a similar manner as has been employed in an investigation of the isoelectronic CH, molecule [S], with which comparison will be made whenever it appears to be appropriate. Only the H-N-H conformations will be studied in the present work, since the N-H-H conformer generally lies higher in energy [4];
The possible formation of the NH: ion from its constituents Nf plus H, and N plus Hi has been investigated in detail in molecular beam experiments [l, 21, and careful theoretical studies of the N + + H, ion-molecule reactions by ab initio calculations have been reported at various levels of sophistication in both the A0 basis set constitution and configuration interaction treatment employed [3-51. The theoretical work finds the ion to be most stable in its symmetric HNH form in the ‘B, electronic state and predicts an intersystem crossing of ‘B, and ‘A, potential surfaces (in a C, perturbation) to play the dominant role in formation of NH: from the NC(3P.J and H,(‘C:) fragments. Further calculations have also considered the formation of NH; from the N atom and the Hi ion as weil as in the outgoing NH(4X-) + H channel via a quintuplet state [6]. 167
168
S.D. Peyerimhofi R.J. Buenker/Potential sur/nces for the lorv-lying stores oJNHl
the various separated species N’, N, H2 and H,C will also be calculated as reference points to be combined with earlier theoretical results for the lowenergy dissociation surfaces of the NH; system
Table 1 Electronic states of NH: for which SCF calculations have
been carried out State No.
2. Technical details of the treatment The A0 basis set employed in the present work consists of the (9s. 5p) set for nitrogen given by Huzindga [9] in the (5s, 3p) contraction recommended by Dunning [lo], in addition to two nitrogen d functions with exponents a,(d,) = 1.8846 and a,(d,) = 0.5582 which have already been employed in an earlier treatment of the Nz molecule [I I]. The hydrogen A0 set consists of a five-term expansion [12] for the 1s function with scaling factor t)’ = 2.0 in a two-term contraction and a one-component representation of a 2p species with exponent a(p,,) = 0.735. Further polarization is taken into account via two bond functions of s and p character each [c&J = 1.8, c&,J =‘0.14, a,(~,) = 1.5, a,(p,) = 0.183 placed in the center of each of the two NH bonds. Thus the entire A0 basis consists of 52 contracted gaussian functions, which are essentially equivalent to those employed in the related treatment of CH, [S]. For a few geometrical conformations additional long-range Rydberg-type functions are also added [n(p) = a(s) = 0.021 in order to ascertain whether such species are important for the higher-energy states under discussion. The configuration interaction treatment is of the general MRD-CI type [13, 141 with configuration selection and energy extrapolation [14]. A core of one orbital corresponding to the Is inner shell is kept doubly occupied in all contigurations and one MO representing the inner-shell complement is disregarded entirely in the treatment, while all other (50) orbitals are allowed variable electron occupation. Self-consistent field solutions are obtained for all the low-energy states given in table 1 which are accessible via the Roothaan method [lo]. These MO’s are generally employed as the basis for the CI calculations, details of which [number of main (or reference) configurations from which single- and double-excitation species are formed, secular equations actually solved (selected configurations)
CorrclationtolinearNH;
Contiguration
WI.
&i, Ibz 3a, Ib, 4a,
.-
2Q 2eU II, 30, -
X’B, $22 I
2
22
1I
1. I
5;
2
z
‘A, 3
2
2
Z
0
‘4
2
2
2
‘A, 4 2 ‘A2 52 ‘B, 6 2
2
0
2
’ “;
2
1
12. 1 1 I 2
2
‘IT,
2
2 I
3
‘%
2
I
3
‘A;72 ‘FL82
I
I
I
I2
z
and configuration spaces for which the CI energy has been extrapolated] are summarized in table 2 for C2, nuclear conformations. All configurations which contribute more than 0.25 “/, (on a c* basis) to the final CI wavefunction expansion are thereby included in the reference set. Finally an estimate of the full CI energy (within the A0 basis set given) is
Table 2 Detaits of the CI treatment” undertaken for the various states of NH,+ State
No.
MO Contiguration basis generation/ selection
Secular equations generated/solved
‘B,
I
3B,
‘Bt ‘A, 2’A, 4lA, 3A, 3B, ‘A, ‘Bz 5A2 ‘AI 23B, 3lA, 2’B,
2 3 4 9 5 6 7 8 10 II 13 14 IS
‘B,
IMlR IMIR
8426/l 933-2624 5454/1528-2277
IA,
8M3R
20034/1866-3177
“A, 3B, ‘A, ‘B2 5A, ‘4
lMlR lM1R 1MlR lM1R 2MlR 3MlR
IA, 3A, 5A,
4M2R 7MlR
8525/1723-2555 757911692-2326 5496/1407-2198 5041/1386-2036 15702/1433-1982 19610/1985-2501 52475/2017-21 I1 14179/2576-3299 31972/1739-2005
7MlR
continued on following page
S.D. Peyeriothofi R.J. Buenker/PotentiaI srr$uces for the low-lying stares of NH; Table 2 (continued) State
No.
Conftgurations (occupation) ?a, IbZ 3a, lb, 4a, 2b, 3b, 2b, Sa,
3,4,9
2 2 2 2 2 2 2 I 2 I 2 2 22-l-l 22-l--1 2 -21--l
2 2 2 2
‘AZ 10
2 1 1 1211---l
1
1
3A,
2
2 2
1 -
1 1
1 -
1
1
1 1 1
1 1 I
1
1
-
1
2 2
1 1 1
‘A,
11
1
=B,
13.15 (1.2)
2 2 2 1 2 2
3lA,
14 (3)
2 2 1 -
2 2 2
1
2 2 2 2
2 2 I
1 1
1
2 2 2 2
2 1
1
1
I 1
1
1
” The term nMmR denotes that n main or reference conligurations are employed for generation of the configuration space (to which the MRD-CI energy is extrapolated) while configuration selection is undertaken for m roots.
169
study obtains for its optimal geometrical parameters NH = 1.940, and L HNH = 150” (CI value’). The calculated ground-state SCF energy (at NH = 1.940775a0, L HNH = 1SS’) is -55.22321 hartree, a value which is very close to that calculated by Bender et al. [3] (-55.22965 hartree at L HNH = 143.3” and NH = 1.9238a,) employing a near Hartree-Fock basis of N (13sSp3d/9s6p3d) and H (6s2p/4s2p) functions. The corresponding CI energy in the present work is - 55.3755 hartree at the MRD-CI level and -55.382 hartree for the full-C1 estimate. The molecule must be considered as quasi-linear in its ‘B, ground state since the barrier (difference of total electronic energies at L HNH = 155” and 180’) to linearity is found to be only 330 cm-’ at the full-C1 limit (60 cm-’ more at the MRD-CI level), while the zero vibrational level for the bending frequency is calculated to be of practically the same magnitude (318 cm-‘, K = 0). The SCF treatment yields a barrier approximately three times as large. Comparison with the calculated energy for the neutral NH, species in its equilibrium (‘B, state, NH = 1.935106a,, L HNH = 103.40”$ yields an energy separation (neglecting vibrational effects) between the X3B, ion and the neutral X’B, species of T, = 10.9 eV, while the calculated urrrical (electronic) energy difference is found to be 11.6 eV (only 10.4 eV at the SCF level of treatment), a value which compares quite satisfactorily with the measured IP of 11.4 eV determined from massspectrometric data [17].
undertaken according to the generalization AE = (1 - Cc&%,r
- L,,,_cr),
of a formula given by Davidson [16] derived for the contribution of higher than double-excitation configurations. In virtually all cases the sum of the squares of coefficients for the reference species Zci is found to lie between 93 % and 96 %_
3. Potential energy surfaces 3.1. The NH: ground stute The ground state of NH: is 3B, with the electronic conliguration la:2aflb$3a, lb, ; the present
3.2. Atzgular potential curves The various angular potential curves obtained from the CI calculations at a fixed NH separation of NH = 1.940775a, for the lowest-energy states of NHf are contained in fig. 1; the calculations are generally carried out for internuclear angles of 180, 165,155,140,125,115,105,90,75, and 60 degrees, and these are complemented in some cases at additional angles of 45” and 30”. The various optimal geometrical parameters obtained from ’ The corresponding SCF calculations yield a somewhat smaller internuclear angle of 145”. ’ E(SCF) = -55.57995 hartree, E(MRD-CI) = -55.77318 hartree and E(full-CI est.) = -55.7839 hartree.
170
S.D. PeJleritnho/j. R.J. BuenkerlPotential surfaces/or the
loiv-lying srares ofNH;
Table 3 Calculated internuclear angles and corresponding energies of NH; obtained from polynomial fits to the curves in lig. 1 (The NH distance is 1.940775no throughout.) State
Energy (eV)
‘B, ‘A,
149.6 107.6 155.2 180 60 63 89 98
I%
-N?o”‘.H; -N’klpi2
2 ‘A, ‘A, ‘A, ‘Bz ‘B, -‘Ai 2’B,
0.0
1.29 2.03 3.45 4.12 5.64 7.42 9.24 Il.09
‘H”
-N&&n; -ril+gl.HI
4 ‘A,
180” Z.50
14.2 zs.5
a’ The calculations also show a second minimum around 55”.
Fig. 1. Angular potential curves for a series of NH: states obtained at the (estimated) full-Cl level of treatment (RNH = 1.940775~~);the energies of various dissociation limits (for optimum Hz and Hi bond lengths, respectively) are also indicated. Energy values are given in units of
hartree, HNH angles in degrees. polynomial fits to the data points of fig. 1 are contained in table 3, together with pertinent energy values. It is obvious that the three lowest states, 3B1, ‘A, and ‘B,, are quite similar in character to the corresponding well-known states in the isoelectronic CH, molecule. The equilibrium angles in NH: are all about 5” larger than in CH, [8] (‘A,, 103”; 3B,, 135”; IB,, 140”) and the angular potential curves are more shallow in NH:, although their relative forms are in close analogy to those found for CH,, including the characteristic that the ‘B, internuclear angle is approximateiy 5” larger than that of the corresponding triplet state. This general similarity between NH: and CHI states has already been predicted on the basis of simple MO theory in a
refined Mulliken-Walsh model [18,12,19]. The splitting between ‘A, and 3B, is calculated to be AE, = 29.9 kcal/mole; this value is considerably larger than for the equivalent energy separation in CH,, which ranges in the most recent literature [S, 20-221 from 8.0 to 11 kcal/mole (although photodetachment measurements [23] yield a value of 19.5 + 0.7 kcal/moleT). A two-configuration description of the *A, state relative to the singleconfiguration (SCF) treatment of the 3B, species yields a value for the singlet-triplet splitting of 29.1 kcal[3] in a near Hartree-Fock A0 basis, demonstrating that the dominant parts of the CI expansions of the two states under discussion can be accurately represented by two and one terms, respectively. Other theoretical ab initio predictions [24-261 range from 36 to 45 kcal, while a semiempirical MIND0/3 treatment [27] yields 31 kcal for this quantity. As in CH, the SCF singleconfiguration treatment overestimates this singlettriplet energy splitting and results in the present A0 1
Since all recent theoretical results for this quantity as well as those obtained in thermochemical studies lie in the 8-l 1 kcal area, the interpretation of the photodetechment experiments [23] is presently under extensive study in various laboratories.
S.D. Pqerintltoff, R.J. Buenker/Potential sudaces for the low-lyingSates of NH:
basis in a value of 44.3 kcal, in close proximity to the 44.7 kcal reported in ref. [3] using an extended A0 basis. The second ‘A, species is found to be linear, and this geometry is also expected for the corresponding state of CH,, which has received relatively little attention so far. The two states correlating with ‘fls and ‘H,, respectively, prefer bent geometries since they depopulate the cU-b2 orbital with its strong preference for the linear arrangement of nuclei [ 193. This situation is especially apparent for the states of A, symmetry with a la:2a:lb,3aflb, electronic conliguration since they populate the 3a, MO (which distinctly prefers the bent conformation [l9]) twice in contrast to the Bz states with laf2a:l b23ar 1b: occupation which have the essentially non-bonding MO lb, doubly occupied. The actual energy dillerence between 3Az at its optimal angle (table 3) and 3Bi at an NH distance of 1.940775a, is 4.12 eV (97 kcal) but, as will be seen in the next section, the former is considerably more stable at larger NH bond distances. This effect is even more enhanced in the 4&A, state, in which the lb, MO is not occupied at all; its potential energy curve shows a deep minimum at very small internuclear angle, and leads to avoided crossings with other states of ‘Al symmetry in energy regions for which this state is quite stable (fig. 1). The higher states 3~‘H,(1tr~20~ 1~~~7r~3tr.Jand 5Z-(lo,22u,Z lo,x,23a,J all populate the 3oe4a, MO (taile 1). Since the 4a, species would be a Rydbergtype orbital in neutral AH, systems, supplementary SCF calculations are carried out (at angle 140”) in a basis containing additional long-range 3s- and 3p-type [U(S) = 0.02, EL(P)= Cl.021functions in order to check whether they make any contribution to the 4ar MO in NH:. Not surprisingly the total energies of the first nine SCF states (table I) are lowered only a small amount (O.tlOOO6 hartree, less than 0.002 eV) as a result of this A0 basis extension, but !arger effects are noted for the higher states which occupy 4a,, namely 0.066 eV, 0.072 eV and 0.154 eV, respectively, for the 5A, (IO), 3B2 (12) and ‘A, (11) states of table 1. Nevertheless, such changes seem small enough to justify omission of Rydberg-type functions for the present investigation. The preferred linear geometry in the 3*rHUstates can also be rationalized in terms of MO theory: the
171
effect of doubly occupying the 1b2 orbital (which is considerably more stable for L HNH = 180” [ 191) cannot be offset with only single occupation of the 3a, MO with its bent tendency, especially since the orbital energy of the 4a, species does not vary greatly with change in internuclear angle. In a similar manner the 5A, state shows an energy minimum for the linear nuclear framework, but with a much smaller bending force constant than for the 3-1HUstates; the latter distinction is clearly traceable to the distinction in the lb, occupation numbers of the 5A, relative to these lower-lying states. At smaller internuclear angles the combined tendency of the 3a, and lb, species, which prefer bent NH, geometries, apparently overcomes the influence of the 1b, MO, producing a second minimum at a very small internuclear angle (fig. 1). 3.3. Sytnmetric NH stretch potential curves The potential energy curves for symmetric stretch are contained in figs. 2a, and 2b; they are calculated at the SCF level of treatment for only two internuclear angles: 140” (close to the X’B, minimum) and 105” (which is near the minimum of the lowest NH: singlet state). Similar to the situation for the angular potential energy curves it is seen that the electronic states can be separated into three major groups according to their occupation of the (3a,, lb,), lb, and 4a, MO’s, respectively.’ From earlier calculations [ 191 it is known that the lb, and 3a, MO’s in AH, systems show approximately the same NH bonding behavior and hence the states 3B,, ‘B,, ‘A, and 2’A, show very little distinction in their equilibrium bond lengths (figs. 2a, b and also table 4, which shows the results of a polynomial fit to the data points of lig. 2). The lb, MO is found to be more bonding [19] than 3a, or 1b, and hence depopulating this MO, as is done in the A2 and B, states, leads to the increase in bond lengths obvious from fg. 2. Finally, occupation of the strongly NH antibonding 4a, orbital produces a very large NH equilibrium separation or even a completely repulsive potential curve; such a fragmentation is already obtained at the SCF levei of treatment for the ‘A2 state, for example, which dissociates into N(%,) -t Hz(‘Z:).
S.D. Peperimhc& R.J. BuenkerjPotential surfaces/or the low-lying states of NH;
172
mar1rL
-5L.70
-5L.80
-54.90
-55.m
-55.10
-55.10
-55.x
- 55.20
i.eb
* 2.04
2.21
2.U
NH
laoI
Fig. 2. Symmetric NH stretch potential curves obtained at the SCF level of treatment for fixed HNH internuclear angles of (a) 105” and(b) 140”; the energy of the most stable N+ + H2 dissociation product is aIso indicated.
3.4. Potentiul curves@
antisyrmetric
stretch
Potential curves for asymmetric stretch are only calculated in the immediate neighborhood of the potential minima obtained thus far, with the goal of ascertaining whether the various states prefer (locally) symmetric nuclear arrangements; in addition, approximate values for the antisymmetric vibrational frequencies in these states are also obtained. The results of these (SCF) calculations are collected in table 5 for a series of electronic states. It is found that the low-energy species 3B,, ‘B,, ‘A,, 2’A, and 3B, are all stabie with respect to such a C, distortion around their respective potential minima. Such is not the case, however, for
the ‘A, state which populates the 4a, MO; the results of table 5 show furthermore that this distortion is somewhat more preferred at small internuclear angles than for nearly linear HNH arrangements. Taken together with the results of the previous section it seems very likely that this state (along with the closely related species 3 ‘A, and 23*1B,) is not a stable entity. The 3A, and ‘A, states show a definite increase in energy for small antisymmetric distortion (AR = 0.1 bohi), but exhibit more dissociative characteristics after passing over this barrier; this behavior is also consistent with results of earlier investigations of the ‘AZ potential surface [3,4].
S.D. Peyerimhofi
R.J. BuenkerjPotential
Table 4 Calculated equilibrium separations of NH: (symmetric arrangement) and corresponding energy differences from the conformation employing R,, = 1.940775a,; the values are obtained from a polynomial tit to the SCF data of tig 2
surfacesfor
2
R,
AE(R-
AE(R - R,)
R,) R,
% ‘A,
1.940
-
1.925
-
1.957
-
1.934
-
‘BI
1.962
-
1.933
-
‘A, 3Az
1.989
0.02 eV
1.943
-
2.210 2.247 2.193 2.205 _b) -
0.56eV 0.71 0.51 0.55 -
IA, 3B* ‘Bz 3A, 5Az 4’A,
a) See fig. 2a.
0.47 0.55 0.39 0.38 >2.80=’ > 4.9 >2.5
-
” See fig. 2b.
3.X Separutel1 products
R(a,)
0
1.940775 1.940775 1.940775 1.940775 1.940775 1.940775
155” I55O 155” 155” 155” 155” 105” 105” lOS0 105” 105” 105” 90”
AR=O.lu,
AR=O.Za,
0 = 140
0 = 105”
2.18 2.20 2.156 2.150 > 2.54 > 2.54 > 2.54
173
Table 5‘ SCF energy changes (in eV) for antisymmetric distortion of the NH,+ framework relative to a given conformation (R, f-J State
State
the low-lying states of NH;
[N
+ H,]
+
The energies of the separated products N+ plus Hz as well as N plus Hl have also been calculated in the present work as reference points for the states of the combined NHf system. In order to provide for a balanced treatment of the combined molecule and the separated fragments, the calculations for the former species are carried out at an internuclear distance of lOa, between the nitrogen nucleus and the H, framework (with the appropriate H-H separations, 2.003a, for Hf and 1_40a, for HJ in an equivalent manner as for NH: near its equilibrium nuclear arrangement. The N(4S,) + Hf(‘Zl) complex can be obtained in Czv symmetry either from a calculation of NH: for a ‘A, or a 3Az multiplet, and the results in table 6 show that there is good numerical agreement in the two technically different calculations; details of the CI treatments are contained in table 7. The corresponding singlet state ‘A, corresponds to N*(‘D,) + H:(*Z:) and is calculated to lie 2.63 eV higher; the experimental N(4S,)-N(2D,) excitation energy is 2.38 eV [28], but it has already been found in an earlier CI study on the dissociation of N, in ground and excited states that the present A0 basis yields a 4SU-*D, splitting which is too large by 0.23 eV.
jB, ‘A,
.‘B, 2 ‘A, gB, 3A, 3B, ‘A, ‘B, 2 ‘A,
3Bz JA* 3BZ
1.940775 1.940775
1.940775 1.940775 1.940775 1.940775 2.140775
0.18 0.19 0.19 0.17 0.23 -0.43 0.11 0.12 0.12 0.10 0.10 -0.54 0.04
0.49 0.49 0.49 0.46 0.49 0.47 _ 0.17
The N+ + Hz fragments can also be obtained in various ways in a formal NH: molecular calculation; details are also given in tables 6 and 7. The ‘B,, 3A, and 3B, states all correlate with the lowest N+(3P,) + H&X:) fragment states and table 6 again shows that there is good numerical agreement in the various treatments when various MO basis sets and configuration spaces are employed. The singlet states ‘B,, ‘A?, and ‘B, of equivalent character as the triplet species correlate with the first excited state of the nitrogen ion, N+(’ D,) + H,(‘Zl). The present calculations lind an energy difference of 2.09 eV for this 1D,-3P, splitting; the corresponding experimental result is 1.89 eV [23].’ Comparison with previous work can also be made at this point. The SCF calculations of Bender et al. [3] yield an N+(3P,) + Hz energy (3A2 state) of - 55.02 123 hartree in a near Hartree-Fock basis and a value of - 55.01159 hartree in their DZ + P basis, while the present work finds an SCF energy of -55.01363 hartree for the same fragments. The CI energy obtained in ref. [3] (employing the vector method) in the DZ + P basis is -55.12329 hartree, recovering 0.1117 hartree of the valence-shell correlation, whereas the present CI energy (MRD-CI level) is -55.13898 hartree, corresponding to a correlation energy of 0.12535 hartree; the total valence-shell correlation energy of the present basis is estimated to be 0.13 18 hartree (full-C1 estimzte for
S.D. Pegeridtofi
174
R.J. BuenkerjPotential surfices for the low-lying states of NH2+
Table 6 Total energies (hartree) obtained for the various [N-H,]’ species at a distance between nitrogen and the Hz framework of lo&,; the correlation with the separated fragments is also indicated State
Calc. No.“’
Fragments
SCF
- 54.99426
MRD-CI
Full CI estimate
- 55.09994 - 55.09975 - 55.0025
-55.1036 -55.1035 - 55.0070
HH = Z.Oa,
5A, 3Az
1
N(4S,) + H:(‘Z;)
‘A2
2 3
N(%,,) •t Hz’@;) N(*D,) + Hf(‘Z,C)
HH = 1.4a. 3&
4
‘& 3A LAS
2 I8
N+(‘PJ N’(3P,) N+(‘D,) N+(‘D,) N+(3P,)
+ H&Y;) -I- H,(‘Z;) + HJ’z;) +c HJ’Z;) H&T)
_ - 54.933101 T 55.013629
-55.13839 -55.14005 -55.06130 - 55.13898 -55.06179
-55.1454 -55.1449 - 55.0685 --55.0683 55.1454
382
9 10 11 12
N+(‘P,) N’(3P,) Ni(‘D,) N’(‘D,)
+ + + +
-54.01359 -54.01359
-55.13900 -55.14005 -55.06157 - 55.06348
-55.1454 -55.1449 -55.068? - 55.0685
IB,
-
H,(‘Z;) H1(‘E;) H,(‘E;) H,(‘E;)
-
‘) Calculation.No. refers to the treatment explained in table 7.
Hz + N’(3P,) is -55.1454 hartree; table 6). The calculations of_Gittins et al. [4] obtain -55.06728 hartree [30] in an essentially DZ basis and the CI treatment of Hirst [SJ yields -55.12846 hartree in a DZ + P A0 basis. Consideration of the results for the X3B, groundstate and the lowest-lying set of fragments Table 7 Details of the CI treatment carried out for the various [N-H?]+ species at a distance between nitrogen and the H, framework (C,, approach) of lOao representing effectively separated fragments (notation as in table 2) State
Calc. No.
MO basis
Configuration generation/ selection
Secular equations solved
5A2 3A, ‘A2
1 2 3
5A, SCF ‘A, SCF ‘A,SCF
2MlR 1MlR 1MlR
15702/ 919 18405/1517 10915/1206
4 5 6 7 8 9 10 II 12
5AzSCF ‘A, SCF ‘A2 SCF ‘A2SCF ‘AZ SCF ‘Bz SCF 3B2 SCF 3B, SCF 3B, SCF
1MlR 2MlR 1MlR IMIR 1MlR 1MlR 2MlR IMIR 2MlR
8526/ 834 15711/ 852 5544f 699 8525f 675 5496J 557 9135/ 638 15711/ 852 6GO6/ 513 10159/ 712
N’CP.J
+ Hr(‘xl) ieads to a dissociation energy of eV; previous theoretical findings for this quantity range from 5.66 eV [30] in a DZ and 5.98 eV in a DZ + P treatment [S]. The potential well for the 3A, state (relative to NC + Hz) is found to be approximately 2.80 eV (65 kcal)‘, which is also slightly deeper than in several previous investigations (1.76 eV [4], 2.63 eV [s] and 60 kcal [3])_ A collection of the most important data pertinent to the correlation between NHf and its [N + Hz]+ fragments is given in table 8 and it is clear from these results as well as those of fig 1 that the correlation diagram given by Fair and Mahan [I], v&h has been widely employed for the interpretation of the mechanism of NH: formation, is in D, = 6.45
need of considerable
‘4 ‘B, 3Az ‘A, =Bz ‘B1
revision
[or quantitative
use.
4. Intensities in the NH: spectrum In order to obtain a more complete picture of the possible absorption (or emission) spectrum of the ’ In this case the energy at equilibrium is determined as the value for the angular minimum obtained from CI calculations at R = 1.94u,, combined wifh the difference in SCF energies for this NH distance and the optimal separation of 2.19ao.
S.D. Peyerinrhojf, R.3. Buenker/Potential sur/acesfor the low-lpirrg states o/NHf
Table 8 Energies (in eV) of the various (bound) NH, states relative to the separated products. The values, unless directly calculated, are estimated from the calculated data in tables 3 and 4, and the corresponding nuclear conformations are given in parentheses State/Configuration N+(3P,) + H#&+) N(%,) -t Hf(“X;) N+(‘DJ + H&X;) N(‘DJ + I&+(*X;,
% ‘A, ‘B, 2 ‘A, 3Az ‘AI 3B2 rBz
A& 0.0 1.14 2.09
3.76 (3.46,see text) -6.45 (IW, 1.94) -5.16 (108”, 1.94) -4.42 (IS, 1.94) -3.02 (ISO”,1.99) -2.79 (60”,2.18) - 1.36(63”,2.20) 0.58 (89q 2.16) 2.41 (98”.2.!6)
NH: ion the intensities for transition between the states in fig. 1 are also of importance. Hence the electronic transition moments andfvalues between the upper triplet states and X3B, are determined at L HNH = 125”, which represents an average equilibrium angle for the states in question+. The transition moments connecting the upper singlet species with the lowest singlet NH; state are determined at L HNH = 105”, since this angle is close to the equilibrium value of the lowest ‘A, state. In all cases the MO’s of the upper state (see table 2) are generally employed in obtaining the ground state (3B, or ‘A,) CI expansion so as to avoid evaluating the respective matrix elements in a non-orthogonal MO basis. For comparison with the triplet results the corresponding singlet transitions have also been calculated at 125”. The resulting intensity data are summarized in table 9, together with the corresponding vertical electronic energies at the given angle (hence this energy is only approximately equal to the vertical excitation energy from the lower-state equilibrium conformation). It is seen that the 3A,-3B, transition is relatively weak and that thefvalue obtained is quite consis-
’ Note that at 180” the interesting %Is-%;
transition is forbidden under the dipole selection rules, and also that the calculated equilibrium angle for ‘B, is 150”.
175
tent in three different MO basis sets. Furthermore, it is clear that the different forms of the 3A, and ‘B, potential curves (fig. 1) will effect a long progression in the bending vibration for this electronic system. As a result the ‘B,A~A, transition is not expected to occur as a strong feature in the NH: spectrum. The 3A1-3B1 transition moment matrix element is quite large (table 9), but since 3A, exhibits a repulsive potential curve along the N-H stretching coordinate a long progression corresponding to a dissociative state is expected in this instance; this transition should most likely be observed experimentally as an almost structureless underlying feature in the NH; spectrum at relatively high energy. Of the remaining triplet-triplet excitations the 3B,-3B, species is not allowed according to the Table 9 CalculatedJvaIues for transitions between various states of NH: states (R = 1.940775a,) Transition
0 = 1X0 ‘AZ-‘B ,
y1
0.00088 0.00077
0.00086 3BL-3B, dipole forbidden 3A,-3BI ‘AZ-‘B, 2 ‘A,-‘B, 3 ‘A,-‘B, 4 ‘A,-‘B, *B,-* B,
0.014 0.001 I 0.0020 0.013 0.55-5 dipole
‘B,-‘A,
0.00025
A&,,, (cV)
6.62
MO’s employed to calculate /value
3A1 MO’s 3B, MO’s
‘A, MO’s 7.72 12.6 5.77 2.30 11.37 14.55 7.19
‘A, MO’s ‘A2 MO’s ‘A, MO’s ‘A, MO’s ‘A, MO’s
0.84
‘A, MO’s
1.66 4.73 5.66
‘BI MO’s ‘A, MO’s
forbidden 0 = 1,050 ‘B,-‘A, 2 IA,-‘A, ‘AZ-IA,
0.00086 0.001 I dipole forbidden ‘B,-‘A, 0.6-6 3 ‘A,-‘A, 0.12 4 ‘A,-‘A, 0.15-4 2 ‘A,-‘B, 0.0038 ‘B2-2 ‘AI 0.0010 3 ‘A,-2 ‘A, 0.00086
7.95 13.2 13.44 3.08 3.23 8.57
‘B, ‘A, IA, ‘Br ‘Bz “A,
MO’s MO’s MO’s MO’s MO’s MO’s
‘t The value of the oscillator strength is calculated
according tof= (2/3)l(Y,/ R,.,,. /Yu,)l’ BE. The notation 0.n - p stands ior 0.n x 10-p.
176
S.D.
Peyeritnizoff, R.J. Buenker/Potential surfaces for the low-lying states of NHf
dipole-selection rules and the 23B,-X3B, counterpart corresponds to an even higher transition energy than that of 3A,-‘B, ; since the latter upper state is also repulsive in the NH stretching coordinate, similar to the 3A,, its transition moment relative to 3B, has not been calculated. The corresponding transition from ‘A, and 3 ‘A, to the ‘B, state are of similar intensities as those between the corresponding triplets (table 9). The additional 21A,-‘B1 intercombination, which does not possess a triplet counterpart, has anfva!ue of 0.0020, but the transition is also forbidden in the linear nuclear geometry; since the shape of both curves is very similar the Franck-Condon factors are expected to be quite large for intercombination of equivalent vibrational states. _ The transitions to the lowest bent singlet state are also relatively weak for all species which have gerade symmetry in the linear molecule. The uttgerade states such as 3’ A, show a largefvalue for combination with ‘A,, but again as in the triplet case discussed earlier the 3 “Al is very high in energy, lying above the lowest dissociation limits for this system, and shows a predominantly repulsive potential curve towards NH stretch. The oscillator strength of the ‘B,-‘A, transition, on the other hand, is ofsimiiar magnitude as the corresponding ‘BI-‘AI band system in NH2 which is well understood, so that both ‘B,-‘A, and 2lA,-‘A, transitions should be relatively easy to observe in addition to 21A,-‘B1 intercombinations if the NH: system can be found in sufficient concentration to afford measurement of its electronic spectrum.
c31C.F. Bender, J.H. Meadows and H.F. Schaefer III, Faraday Discussions Chem. Sot. 62 (1977) 59. [41 M.A. Gittins, D.M. Hirst and M.F. Guest, Faraday
Discussions Chem. Sot. 62 (1977) 67; M.k Gittins and D.M. Hirst. Chem. Phys. Letters 35 (1975) 534. D.M. Hirst, Mol. Phys. 35 (1978) 1559. ;i; D.M. Hirst, Chem. Phys. Letters 53 (1978) 125. c71G. Herzberg Quart. Rev. 15 (1971) 201; and private communication. PI S. Shih, SD. Peyerimhoff and R.J. Buenker, Chem. Phys. Letters 55 (1978) 206.
PI S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. T-H. Dunning, J. Chem. Phys. 53 (1970) 2823.
;t:; W. Butscher, S. Shih, R.J. Buenker and ?.D. Peyerimhoff, Chem. Phys. Letters 52 (1977) 457.
Cl21S.D. Peyerimhoff, R.J. Buenker and L.C. Allen, J. Chem. Phys. 45 (1966) 734, Appendix.
Cl31R.J. Buenker, S.D. Peyerimhoff and W. Butscher, Mol. Phys. 35 (1978) 771.
Cl43R.J. Buenker and S.D. Peyerimhoff, Theoret. Chim. Acta 35 (1974) 33; 39 (1975) 217.
EN C.C.J. Roothaan, Rev. Mod. Phys. 32 (1960) 179. E.R. Davidson, in: The world of quantum chemistry, II161 eds. R. Daudel and B. Pullmann (Reidel, Dordrecht, 1974) p. 17; S.R. Langhoff and E.R. Davidson, Int. J. Quantum Chem. 8 (1974) 61. Cl71G. Herzberg, Electronic spectra of polyatomic molecules (Van Nostrand, New York, 1966).
Cl81A.D. Waish, J. Chem. Sot. (1953)2260. Cl91R.J. Buenker and SD. Peyerimhoff, Chem. Rev. 74 (1974) 127.
c201R.K. Lengel and R.N. Zare, J. Am. Chem. Sot. 100 (1978) 7495, and references therein. P11 L.B. Harding and W.A. Goddard, Chem. Phys. Letfers 55 (1978) 217.
[22] C.W. Bauschlicher Jr. and I. Shavitt, J. Am. Chem. Sot. 100 (1978) 739.
c231P.F. Zittel, G.B. Ellison, S.V. O’Neil, E. Herbst, W.C. Lineberger and W.P. Reinhardt, J. Am. Chem. Sot. 98 (1976) 3731.
c241S.T. Lee and K. Morokuma. J. Am. Chem. Sot. 93
Acknowledgement
(1971) 6863.
The authors wish to thank the Computing Center of the University of Bonn for services and computer time made availabIe for this study.
cw S.Y. Chu, A.K.Q. Siu and E.F. Hayes, I. Am. Chem. sot. 94 (1972) 2929.
IPI J.F. Harrison and C.W. Eakers. J. Am. Chem. Sot. 95 (1973) 3467.
c271M.J.S. Dewar, R.C. Haddon, W.K. Li, W. Theil and
[=I
References [I] I.A. Fair and B.H. Mahan, J. Chem. Phys. 62 (1975) 515.
[2] D.J. McCIure, C.H. Dou&dss Chem. Phys. 66 (1977) 2079.
and W.R. Gentry, J.
P-K. Weiner, J. Am. Chem. Sot. 97 (1975) 4540. C. Moore, Atomic energy levels, Nat. Bureau of
Standard Circular No. 467 (1949). W. Butscher, S.-K_ Shih, R.J. Buenker and S.D. Peyerimholf, Chem. Phys. Letters 52 (1977) 457. D.M. Hirst, in: General discussion on potential L-301 energy surfaces, Faraday Discussions Chem. Sot. 62 (1977) 138.
m