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Electronic states and nature of bonding in the molecule RhN by all-electron ab initio calculations
THEO CHEM ELSEVIER
Journal of Molecular Structure (Theochem) 393 (I 997) 127- 139
Electronic states and nature of bonding in the molecule RhN by all-electron ab initio calculations Irene Shim”, Kim Mandixb, Karl A. Gingerich”%* “Department bDepartment
of Applied
of Applied
Chemistry,
Chemistry,
The Technical DK2800
‘Department
Univrrsir_v of Denmark,
ofElectrophysics,
DTLJ 375 and Department of Chemistry,
DTU 375, DK2800
Lyngby,
DTU 322, The Technical
Denmark
University
of Denmark,
Lyngby, Denmurk
Texas A&M
University,
Received 5 June 1996; accepted
College Station,
TX 77843, USA
25 June 1996
Abstract In the present work, all-electron ab initio multi-configuration self-consistent-field (CASSCF) and multi-reference configuration interaction (MRCI) calculations have been carried out to determine the low-lying electronic states of the molecule RhN. In addition, the relativistic corrections for the one-electron Darwin contact term and the relativistic mass-velocity correction have been determined in perturbation calculations. The spectroscopic constants for the seven low-lying electronic states have been derived by solving the SchrGdinger equation for the nuclear motion numerically. The predicted ground state of RhN is ‘C+, and this state is separated from the states 3rI, ‘II, ‘A, 3Cm,‘A and ‘A by transition energies of 1833,4278,6579,8042,9632, and I3 886 cm-‘, respectively. For the ‘C+ ground state, the equilibrium distance has been determined as 1.640 A, and the vibrational frequency as 846 cm-‘. The chemical bond in the ‘X + electronic ground state has triple bond character due to the formation of delocalized bonding a and c orbitals. The chemical bond in the RhN molecule is polar with charge transfer from Rh to N giving rise to a dipole moment of 2.08 Debye at 3.1 a.u. in the ‘C+ ground state. An approximate treatment of the spin-orbit coupling effect shows that the lowest-lying spin-orbit coupled state is O+. This state is essentially derived from the ‘2 + ground state. The second and third state, O+ and O-, mainly arise from the ‘II state. The dissociation energy of the RhN molecule in its ‘C’ ground state has been derived as I .74 eV. 0 1997 Elsevier Science B.V. Keywords:
Metal nitride; Excited state; Potential energy curve; CASSCF calculation;
1. Introduction The fixation of nitrogen in biological as well as in industrial processes requires transition metal catalysts. Therefore, basic knowledge of the chemical bonds between the nitrogen and the transition metal atoms is important for the understanding of the catalytic actions. In view of this, we have initiated
* Corresponding author.
effect
investigations of second row transition metal diatomic nitrides to elucidate their electronic structures and nature of bonding. The first molecule we have investigated is YN [ 11. Such investigations complement our previous work involving the corresponding diatomic carbides [2-71, and they will also permit comparisons of the chemical bonds and of the electronic structures of the transition metal nitrides and carbides. In the present investigation we present results of ab initio calculations for the molecule RhN. The
0166-1280/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved SO 166- 1280(96)04802-6
PII
Relativistic
128
I. Shim et al.Nournd
of Molecular
low-lying states of the RhN molecule have been studied by performing all-electron ab initio HartreeFock (HF), multi configuration self-consistent-field (CASSCF) calculations, and also multi-reference configuration interaction calculations (MRCI) that include single and double excitations from selected reference configurations. The HF calculations have been carried out in the Hartree-Fock-Roothaan formalism [8]. The integrals have been computed using the program MOLECULE [9], and for the HF calculations we have utilized the ALCHEMY program system [lo]. The multi configuration self-consistentheld calculations have been performed using the CASSCF program ([ 11-141, and the multi reference configuration interaction calculations (MRCI) have been carried out using the program system MOLCAS, version 2 [ 151. Furthermore, the relativistic effects on the low-lying electronic states of the RhN molecule have been taken into account by performing perturbation calculations of the one-electron Darwin contact term and of the relativistic mass-velocity correction. Finally, the effect of the spin-orbital coupling on the low-lying states has been considered by carrying out perturbation treatments.
2. Basis sets and HF calculations
on RhN
The basis sets consisted of contracted Gaussiantype functions. For the Rh atom, we have used a basis set that is identical to that used in our previous work on the molecule RhC [3]. It is essentially Huzinaga’s (17s 1 lp, 8d) basis [16], but it has been extended by addition of two p functions with exponents 0.20 and 0.09. These functions are needed to represent the 5p orbitals. In addition, the most diffuse s functions were contracted slightly by altering the exponents from 0.10732168 and 0.03864044 to 0.125 and 0.05, respectively. The primitive basis set (17s 13p, 8d) has been contracted to (10s 8p, 5d) using a segmented contraction scheme. In the contracted basis, the 4d orbital is represented by a triple-zeta function while all other orbitals including the unoccupied 5p orbital are represented by doublezeta functions. For the N atom we have used Huzinaga’s (10s 6p) basis [ 171, but it was augmented by a d polarization function with exponent 0.95. The basis set for the N atom has been contracted to (4s 3p, Id)
Structure (Theochrm) 393 (1997) 127-139
resulting in double-zeta representation of the s functions, triple-zeta representation of the 2p function, and a d polarization function. The above basis sets have been utilized in the major part of our calculations, but in addition we have performed further investigations using a basis set for the Rh atom that has been supplemented by an f function with exponent 1.125. In our previous work on the RhC molecule [3], we have investigated the electronic states of RhC by performing HF and valence configuration interaction (VCI) calculations. The electronic ground state and first excited state of the RhC molecule were determined to be ‘C+ and ‘II with the approximate valence shell configurations (lOc~)‘( 1 la)*( 12a)‘(57r)‘(26)’ and (lOa)*( 1la)*(5~)‘(6~)‘(26)‘, respectively. These findings are in agreement with the available spectroscopic data on the RhC molecule [ 181. Furthermore, in the HF approximation both states, ‘C+ and ?I, were found to be bound relative to the free HF atoms. In addition, it was realized that the low-lying electronic states of the RhC molecule are consistent with a simple molecular orbital diagram [6,19]. In view of the results obtained for the RhC molecule the first assumption as to the electronic ground state for the molecule RhN is ‘C+. This state arises if the additional electron in the RhN molecule, as compared to the RhC molecule, according to the build-up principle, completes the half-filled (Torbital of RhC. In Table 1 we present results of HF calculations for this ‘C+ state as well as for other states that possibly occur when the ‘S, N atom interacts with the Rh atom in either its 4F, (4d)s(5s)’ ground term or in its ID, (4d)” lowest-lying excited term. From the results reported in Table 1, it is noted that there is considerable charge transfer from the Rh to the N atom in all the states investigated. This is in accordance with the difference in electronegativities of the atoms Rh and N. More important, however, it is seen that none of the states shown are bound in the HF approximation relative to the free HF atoms. The lowest lying state identified is ‘A, and this state is unbound by 0.33 eV at the internuclear distance of 3.6 a.u. The ‘C+ state that is the first choice as a likely ground state candidate is unbound by 5.36 eV. These findings differ completely from those of the RhC molecule, where the two lowest-lying states were bound in the HF approximation. The reason for this
1. Shim et al./Journal
of Molecular
Structure
(Theochem)
393 (1997)
Table 1 Total energies for the RhN molecule as resulting from HF calculations at the internuclear moments, the gross atomic charges, and also the number of s and d electrons on Rh State
Valence shell configuration
Energy”/a.u
129
127-139
distance of 3.6 au. Also included are the dipole
Gross atomic Total number charge on Rh of d electrons
Occupation
of Rh orbitals
Dipole moment (Debye)
‘A 3c2SC+ ?rI ?I,* ‘Et % ‘A ‘A
IOU
110
120
130
5?r
6a
26
2
2 2 2 2 2 2 2 2 2
I 0 2 1 2 2 2 2 2
0 0 0 0 0 0 I 1
4 4 4 4 4 4 4 4 4
2 2 2 1
3 4 2 4 3 4 2 3 1
2 2 2 2 2 2 2
I
I 0 1 0 2
0.012148 0.043356 0.082972 0.117730 0.128343 0.196843 0.203752 0.263073 0.3 13739
0.49 0.45 0.48 0.30 0.25 0.41 0.11 0.16 0.49
18.09 18.41 17.64 18.47 18.20 18.20 17.49 17.52 16.83
4du
4dr
4ds
5s
1.15 0.48 1.69 1.03 1.69 1.85 1.80 1.88 1.85
3.93 3.93 3.95 3.43 3.52 2.35 3.79 2.64 3.98
3.00 4.00 2.00 4.00 3.00 4.00 2.00 3.00 1.00
0.32 0.06 0.78 0.16 0.47 0.28
1.09 0.96 1.27
5.48 5.1 I 5.28 3.48 3.51 5.22 1.54 1.23 2.09
’ Energy of RhN minus energy of the Rh 4F(4d)8(5s)’ and the N ‘S(2s)‘(2p)‘.
is connected to the charge transfer. In the RhC molecule the charge transferred from the Rh to the C atom could enter an empty orbital on C. In the RhN molecule, the charge transferred from Rh to N has to go into an orbital on N that is already partly occupied, and therefore the atomic correlation is larger for N as part of the RhN molecule than for the free N atom. This reveals itself as unbound states of the RhN molecule in the HF approximation. Thus, the HF approximation is not adequate for describing the electronic structure of the RhN molecule. Further investigations of the low-lying electronic states of the RhN molecule have been performed by carrying out MCSCF calculations within the framework of CASSCF.
3. Results of CASSCF calculations molecule
on the RhN
In the CASSCF calculations, the core orbitals, i.e. the Is, 2s, 3s, 4s, 2p, 3p, 4p, and 3d of Rh and the 1s orbital of N, were kept fully occupied, while the valence orbitals 5s and 4d of Rh and 2s and 2p of N, have been included in the active space. The CASSCF calculations have been performed in the subgroup CzV of the full symmetry group C,, of the RhN molecule. The calculations have been carried out for singlet, triplet, and quintet states. The number of configurations included in the CASSCF calculations
reached 1308 for the singlet states, 1746 for the triplet states, and 590 for the quintet states. The CASSCF calculations have been performed as functions of the internuclear distance, i.e. for the distances 2.9, 3.0, 3.1, 3.22, 3.28, 3.3, 3.6, 4.2, 5.0, and 12.0 a.u. The potential energies derived in the CASSCF calculations were utilized to solve the Schr iidinger equation for the nuclear motion numerically. On the basis of the results obtained, the electronic ground state of the RhN molecule is derived as ?I. This state is separated from the next higher-lying states 3Cm, ‘C+, ‘II, ‘A, ’ A, ani ‘A, by the calculated transition energies 12 11, 1475, 2130, 3809, 6479, and 7249 cm-‘, respectively. For all the states investigated, we have utilized the orbitals optimized in the CASSCF calculations at the internuclear distance of 3.3 a.u. to perform valence CI calculations. These calculations have shown that there are no additional low-lying electronic states between the states determined in the CASSCF calculations. By applying the build-up principle to the molecular orbital diagram resulting from our previous investigations of the RhC molecule, we expected the RhN molecule to have a ‘C’ electronic ground state. In view of the rather small energy separation of 1475 cm-’ between this ‘C’ state and the predicted ‘II electronic ground state, we decided to perform further investigations to ascertain the symmetry of the electronic ground state of the RhN molecule. In this connection we performed CASSCF calculations using a basis set for the Rh atom that included an f function.
I. Shim et al./Journal
130
of MolecularStructure
In addition, we also carried out more elaborate CI calculations as described in the next section. An f function with exponent 1.125 was added to the Rh basis set, and CASSCF calculations analogous to those described above were performed for the ‘C’ state at the internuclear distances 2.9, 3.0, 3.1, 3.22, 3.28, 3.3, and 3.6 a.u. Equivalent calculations were also carried out on the states “II, ‘C- and ‘II at the internuclear distances 3.0, 3.1, 3.22, 3.3, and 3.6 a.u. The potential energies derived in the CASSCF calculations were utilized to solve the Schrodinger equation for the nuclear motion numerically. The results of the calculations performed revealed that the sequence of the low-lying states does not change when an f function is added to the Rh basis set. The transition energies between the 311 state and the higher-lying ’ -, ‘E+, and ‘II, are derived as 1618, 1741, states -C and 2356 cm-’ respectively. Thus, the inclusion of the f function in the Rh basis set does not change the symmetry of the ground state, rather it enhances the separation between the lowest-lying states. Table 2 shows the spectroscopic constants obtained for the low-lying states of the RhN molecule derived on the basis of the results obtained in the CASSCF calculations, excluding and including the f function in the Rh basis set. Table 2 shows that the effect of adding the f function to the Rh basis set is reduction of the equilibrium distances and, except for the ‘.Z+ state, increase of the
(Theochem) 393 (1997) 127-139
vibrational frequencies. For the four states investigated, “II, ‘Cm, ‘C’ and ‘II, the decrease in the equilibrium distances ranges from 0.015 to 0.031 A. The vibrational frequencies increase by 67 cm-’ for the “II and ‘II states, and by 8 cm-’ for the 3Cmstate, while the vibrational frequency practically does not change for the ‘Cf state.
4. Results of MRCI calculations and ‘C’ of RhN
on the states 311
Using the Rh basis set without the f function, MRCI calculations have been performed on the states 311and ‘Cf of the RhN molecule at the internuclear distance of 3.3 a.u. In these calculations, all core orbitals have been kept frozen. Single and double excitations from the selected reference configurations have been allowed. The reference configurations were generated by keeping the 26 and the two lowest-lying valence e orbitals, 10~ and 1 lo, fully occupied while all rearrangements of six electrons within the orbitals 12a, 5?r, and 6a have been included. This resulted in 55 reference configurations for the ‘C+ state, and in 48 reference configurations for the 311state. In the final calculations, the 18 highest-lying virtual orbitals have been deleted, while all single and double excitations of all the 14 valence electrons in the reference configurations into the 36 remaining virtual
Table 2 Spectroscopic constants of the low-lying electronic states of the RhN molecule as derived from the results of CASSCF calculations sets for the Rh atom excluding and including an f function. Also shown is the population of the 5s orbital of Rh at the internuclear 3.1 a.u. State
Rh basis without f function Equilibrium distance/A
VI ?c ‘c+ ‘n TA ‘A ‘Ab ‘n
I .705 I.770 1.732 1.717 1.873 2.032 I.162 Repulsive
Vibrational frequencykm~’ 794 791 692 752 564 341 819
using basis distance of
Rh basis with an f function Transition energy/cm-’ 0 I149 1413 2360 3742 7304 9184
Dissociation energy”/eV 1.14 0.99 0.96 0.84 0.67 0.35 0.23
Population of Rh 5s at 3.1 au 0.7 1 0.05
I .oo 0.75 0.72 0.76 0.05 0.68
’ Derived as the difference between the total molecular energy at the equilibrium state. b The values shown refer to the inner minimum of the ‘A state.
Equilibrium distance/i 1.675 1.755 1.710 1.686
Vibrational frequency/cm-’
Transition energy/cm-’
861 799 691 819
0 1618 1741 2356
distance and at the internuclear
dislance of 12 a.u. for each
I. Shim et al./Journal
of Molecular Structure (Theochem) 393 (1997) 127-139
orbitals have been included. This resulted in a total of 495,727 configurations for the ‘C’ state and in 697,744 configurations for the 311 state. Relative to the CASSCF results, the gain in correlation energy is 0.198453 a.u. for the ‘C’ state, and 0.194448 a.u. for the 311 state. Although the correlation energy gained for the ‘C’ state is larger than that for the “II state, the energy of the 311state is still lower than that of the ‘C’ state, but only by 0.001976 a.u. = 434 cm-‘. In view of this small energy difference, we decided to carry out further investigations to determine the influence of the relativistic corrections, i.e. the one-electron Darwin contact term and the relativistic mass-velocity correction. In addition, we have also considered the influence of the spin-orbit coupling on the low-lying electronic states.
5. Relativistic RhN
corrections
of the low-lying
states of
Perturbation calculations have been performed to determine the corrections due to the one-electron Darwin contact term and to the relativistic mass-velocity term. The calculations have been performed using both the basis sets, i.e. the one that excludes and the one that includes the f function for Rh. The resulting perturbation energies are added to the original CASSCF energies, and the resulting potential energies
131
have been utilized to solve the SchrBdinger equation for the nuclesr motion numerically. When the relativistic corrections are taken into account, the sequence of the low-lying electronic states are interchanged, resulting in a ‘C+ electronic ground state that is separated from the higher-lying states, ‘II, ‘II, 5A, ‘C-, ‘A and ‘A, by the transition energies 1832, 4278, 6441, b036, 9635, and 13 888 cm-‘, respectively. Thus, the relativistic corrections cause an interchange of the electronic states and also an enlargement of the energy splittings between the individual states. Table 3 shows the spectroscopic constants obtained for the low-lying states of the RhN molecule derived on the basis of the results obtained in the CASSCF calculations including the relativistic corrections. Fig. 1 shows the potential energy curves for the RhN molecule as resulting from the CASSCF calculations including the relativistic corrections. By comparing Tables 2 and 3, it is noted that there is some correlation between the population of the Rh 5s orbital and the relative magnitudes of the relativistic corrections. Thus, the ‘C’ ground state that exhibits the largest relative relativistic correction has a population of l.OOe in the Rh 5s orbital at 3.1 a.u. The states “II, ‘II, ‘A have populations of 0.71, 0.75, and 0.76e, respectively, in the 5s orbital of Rh at 3.1 a.u. The relativistic corrections for these states cause similar adjustments, but less than that of the ‘C’
Table 3 Spectroscopic constants of the low-lying electronic states of the RhN molecule as derived from the results of CASSCF calculations including the relativistic corrections for the one-electron Darwin contact term and the relativistic mass-velocity correction, using bask sets for the Rh atom excluding and including an f function State
Rh basis without f function Equilibrium distance/i
‘c+ ‘n ‘n ‘A
1c
‘A ‘Ah YI
I h.57 I.676 1.698 I.795 1.757 I.946 1.751 Repulsive
Vibrational frequency/cm-’ 843 910 837 749 817 404 832
Rh basis with an f function Transition energy/cm-’ 0 1833 4278 6579 8042 9632 13886
Dissociation energy”/eV
Equilibrium distance/i
I .I4
I.640 I.657 I.674
I .5 I
Vibrational frequency/cm-’ 846 993 925
Transition energy/cm-’ 0 1752 4319
0.39 0.02
a Derived as the difference between the total molecular energy at the equilibrium state. b The values shown refer to the inner minimum of the ‘A state.
distance and at the internuclear
distance of 12 a.u. for each
132
I. Shim et al./Jourtud
of Moieculur Structure
Energy (a.u.) +4832 a.u. -0,84
-0,86
-0,88
-0,90
-0,92
-0,94
-0,96
-0,98 2
I
I
/
I
3
4
5
6
7
I
I
8
9
R (a.u.) Fig. I, Potential energy curves of 8 low-lying electronic states of the RhN molecule as derived from CASSCF calculations including the relativistic corrections for the one-electron Darwin contact term and also the correction for the relativistic mass-velocity term.
state. Therefore, the states 311and ‘C+ are interchanged resulting in the prediction that the ‘C+ state is the ground state of the RhN molecule. The states 3Cmand ‘A only have a population of 0.05e in the 5s orbital of Rh. For both these states the relativistic corrections are much smaller than for the other low-lying electronic states. Therefore these states are located at much higher energies, relative to the ground state, when the relativistic corrections are taken into account. Comparisons between Tables 2 and 3 also show that the equilibrium distances are decreased and the vibrational frequencies are increased when the relativistic corrections are taken into account. The decrease in the equilibrium distances ranges from 0.011 to 0.086 A. The vibrational frequencies are increased up to 185 cm-‘. The effect of the spin-orbit coupling on the
(Theochem) 393 (1997) 127m/39
low-lying electronic states of the RhN molecule has been investigated by carrying out an approximate treatment using the perturbation Hamiltonian H’ = Ci ~(,,)7i.~i. In addition to the diagonal spin-orbit coupling we have also considered the off-diagonal mixing between the triplet states, ‘II and 3Cm, and the singlet states, ‘II, ‘C’, and ‘A. The effect of the spin-orbit coupling is dominated by the Rh atom. Therefore, we have utilized the atomic spin-orbit coupling constant for Rh together with the atomic population on this atom as derived in the Mulliken population analyses to obtain the splittings between the molecular states. The radial integrals arising when evaluating the matrix elements of the spin-orbit coupling operator are all assumed to be equal, and the value 992 cm-’ has been obtained on the basis of the atomic spectra in Moore’s tables [ 201. The perturbation treatment has been carried out for the internuclear distances 3.0,3.1, 3.22, 3.28, 3.3, and 3.6 au. The potential energies derived have been utilized to solve the Schrodinger equation for the nuclear motion numerically. The resulting spectroscopic constants are presented in Table 4. Since the population in the open shells has only approximately 30% Rh character, the splittings between the spin-orbit coupled components are relative small. The three lowestlying states all have .I; = 0; thereafter follows a state with fl = 2, three states with n = 1, and finally a state with Q = 2. The ground state is essentially the ‘Cm state, but with some admixture of the ‘II0 state. The second and third states, both with Jz = 0, are essentially the “II,+ and ‘ITo- states. Thereafter follows a state with Q = 2. This state is mainly the ‘II2 state, but with a small admixture of the ‘A2 state. Thereafter follows three states all with Jr = 1. These states are mixtures of the states 3C-, ‘II and ‘II. Further details are presented in Table 4.
6. The low-lying molecule
electronic
states of the RhN
Table 3 shows that the energy splittings between the low-lying electronic states of RhN are not significantly influenced by the f function in the Rh basis set. Therefore the following discussion is based on the results obtained with the basis set that does not include an f function.
I. Shim et al.Nournd
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Structure
(Theochem)
393 (1997)
133
127-139
Table 4 Spectroscopic constants of the low-lying spin-orbit coupled electronic states of the RhN molecule as derived from the results of CASSCF calculations including the relativistic corrections. i.e. the one-electron Darwin contact and the mass-velocity, using the basis set for the Rh atom that does not include an f function. Also shown are the contributions of the non-spin-orbit coupled states to each spin-orbit coupled state at the internuclear distance of 3.22 a.u. Equilibrium distance/A
State
Vibrational fequencylcm-
Transition energy/cm-’
D
Contribution ‘C’
0+
1.660 1.674 1.873 1.674 1.683 1.684 1.757
0’ O_ 2 I
I 1 2
1.75I
0
901 957 955 956 900 899 817 831
8 92 100 100 SO 50
92 8
I825 1972 2112 3074 3360 8223 14062
50 50
From Table 5, it is recognized that the major configuration in the ‘Cf ground state at internuclear distances shorter than 3.6 a.u. is the configuration (lOa)*( 1 la)‘( 12a)2(57r)4(2&)4, but with some admixture of the configuration (lOa)‘(l lt)‘(( 120)5a)~(6n)’ (26)“. Table 6 shows that this gives rise to a population of 0.40e in the 6a orbitals at 3.3 a.u. The configuration (10a)‘(lla)2(12a)2(51r)1(2~)4 is the one expected if the build-up principle is applied to the
the low-lying electronic states,
Contribution Internuclear
‘c+
‘n
3YE -
‘A
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
120
2 2 2 2 2 2 2 2 2 2 2 0 1 2 2
I 0
130
5a
2 2 2
0 0 0
I
1
2 0
0 2 I 0 0 0 0 0 0 0 0
I 1 1 1 0 2 1 0 I
I 1
1 2
‘A
100
Valence shell configuration
I10
3c
100
Table 5 The major contributions to the CASSCF wave functions describing functions of the internuclear distance
100
coupled states
‘n
TT
Table 5 shows the contributions of the major configurations to the CASSCF wave functions for four low-lying electronic states, ‘C’, ‘IT, 31T, and, ‘A, as functions of the internuclear distance. Table 6 shows the Mulliken population analyses as derived from the CASSCF wave functions for the ‘C+ electronic ground state as well as for the low-lying excited states, ‘II and ‘Cm, at the internuclear distance 3.3 a.u.
State
(‘%) of non-spin-orbit
67r 4 4 2 2 3 3
I 4 2 3 4 4 4 4 4 4 4
2.9
26 0 2 2 2 1 1 3 1 3 2 2 2 2 2 2 2 2
4 2 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3
’C ‘, ‘II, ‘Cm,and ‘A. of the RhN molecule as
of valence shell configuration
(%)
distance/au.
3.1
3.3
86
83
8
9
3.6
79 I 12
4.2
69 1 19
5.0
38 3 34
I
9 39 13 23 81 6 5
7 6 21 6 17 7 6 8 6 41 60 19 21
88 4 6
82 10 6
68 I9 13
3
89 4
1 87 5
2 83 6
90 2
88 3
89
85
I
3 77 9 2 85 3
1. Shim et al./Journal ofMolecular Structure (Theochem) 393 (1997) 127-139
134 Population
Rh,4d6
Rh.4dTi
2
---__
1
0 t 2
--m r
I
6
gq
T-~-
7
8
R/a.u.
Overlap Population
0.8;
0.6:
0.4;
molecular orbital diagram resulting from our previous investigations of the RhC molecule, i.e. if the additional electron in RhN as compared to RhC completes the half-filled 12a orbital. At the respective equilibrium distances, the low-lying excited electronic states essentially arise from the configurations (10a)2(11a)2(12a)‘(5~)3(6*)‘(26)1, the state ‘II, the states 5A and “A, and ( ~OIJ)~(1l~)2(5~)‘(67r)‘(2s)“, the states ‘C- and ‘A. In all the low-lying states, charge is transferred from the 5n to the 67r orbitals thus contributing to the radial correlation of the 7~ electrons. In the case of the %I state, the singly occupied u and ?r orbitals are triplet-coupled while they are singlet-coupled in the ‘A state. Likewise, in the states ‘C- and ‘A the two electrons in the 6a orbitals are triplet-coupled, while they are singlet-coupled in the ‘A state. The state ‘A consists of configurations in which the spins of the electrons in the 6n orbitals are singlet- as well as triplet-coupled. In accordance with Hund’s rule, the state with the highest spin angular momentum within a given configuration has the lowest energy. Table 5 shows that the state ‘A changes configuration between 3.3 and 3.6 au. The leading configuration in the ‘A state is (IOCT)~( 1 la)‘(5?r)J(6a)2(26)3 for internuclear distances of 3.3 a.u. or less. At internuclear distances 3.6 au. and larger, the configuration has changed to (lOa)‘(l Iaf2( 12a)‘(57r)‘(6?r)*(2&)‘. As seen in Fig. 1, this change of configuration reveals itself as a double minimum in the potential energy curve for the ‘A state. Fig. 2a, Fig. 3a, and Fig. 4a present the analyses of the molecular orbitals for the ‘C’ ground state and for the low-lying excited states, %I and “C-, as populations of the individual atomic orbitals and as functions of the internuclear distance. Fig. 2b, Fig. 3b and Fig. 4b show the total overlap populations as well as the overlap populations due to the valence u and r orbitals for the states, ‘C+, “II, and 3Cm,as functions of the internuclear distance.
0.2-
i 0.0
1 2
3
a
Tsq
I
4
5 R/a.u.
6
7
8
Fig. 2. a. Populations associated with the valence orbitals of the atoms Rh and N in the ‘C+ electronic ground state of RhN as derived from CASSCF wave functions. The equilibrium distance of the state is indicated by rrq. b. Total overlap populations and also overlap populations associated with the valence (r and ?r orbitals in the ‘C’ electronic ground state of RhN as derived from CASSCF wave functions. The equilibrium distance of the state is indicated by req.
1. Shim et c~l.Nournul ofMoleculur Population 47
\ ‘\
‘\
,Rh,4dd .‘-..
‘. ‘1
Rh,4dn 3i
2
1
c
,/c-.
I Rh,4du /-
,/
1 1I
0
~ 2
3T
I
A
r
4
5
eq
R/a.u.
Overlap Population 0.8
0.6
.*x
Ti
t
total
t
6
7
8
Structure (Theochrm)
135
393 (IYY7) 127-139
From Table 6, it is recognized that the I Oa orbital of all three low-lying states is essentially the non-bonding N 2s orbital. The 1 la orbital is the bonding combination of the Rh 4da and the N 2pa orbitals. In the case of the ‘C’ state, the 1 la orbital also has some admixture of the Rh 5s orbital. The 120 orbital is an essentially non-bonding orbital located mainly on the Rh atom. In the states ‘C’ and ‘II, the 12~ orbital has a large contribution from the Rh 5s orbital. The polarization of the 120 orbital results in removal of charge from the internuclear region. The 26 orbitals are the non-bonding Rh 4d6 orbitals. At 3.3 a.u., the Rh 4d6 orbitals are practically fully occupied for the three states, ‘C’, ’II, and ‘Cm, considered. The 5a and 6a orbitals are conventional bonding and anti-bonding combinations of the Rh 4d7r and the N 2p7r orbitals. Table 6 also shows that in all three states there is charge transfer from the Rh to the N atom resulting in positive gross atomic charges on Rh of 0.36e, 0.35e, and 0.46e at 3.3 a.u. in the states ‘C+, ‘7r, and ‘C-, respectively. Furthermore, it is noted that the population of the 5s orbital of Rh at 3.3 a.u. amounts to 0.87e, 0.64e, and 0.05e in the states ‘C’, ‘II, and ‘Cm, respectively. This indicates that the states ‘C+ and ‘II can both be considered as derived from the ‘F, (4d)‘(5s)’ ground term of the Rh atom, whereas the ‘C- state is derived from the ‘D, (4d)’ excited term of Rh. This finding is also consistent with the fact that coupling of the angular momenta of the ‘F, ground term of Rh with those of the ‘S, ground term of N gives rise to singlet, triplet, quintet, and septet states of C’, II, A, and @ symmetry, while the angular momenta of the ‘D, and the %i Uterms couple to triplet and quintet states of the symmetries Cm, II, and A. The results of the Mulliken population analysis indicate that the chemical bond in the ‘C+ electronic ground state of the RhN molecule has triple bond character due to the formation of the delocalized K and u orbitals. The sequence of the low-lying electronic states of the RhN molecule is consistent with a qualitative Fig. 3. Populations
associated with the valence orbitals of the atoms
Rh and N in the ‘fI
low-lymg
from CASSCF wave functions.
electronic
state of RhN as derived
The equilibrium
distance of the state
is indicated
by T_,. b. Total overlap populations
populations
associated with the valence o and ?r orbitals
low-lying
electromc
state of RhN as derived
functions.
The equilibrium
and also overlap in the ‘fl
from CASSCF
distance of the state IS indicated
wave by rc4.
1. Shim et ul.Nournal
136 Population
of MolecularStructure
Rh,4dd
t-
4-
:
Ijl
A
Rh,4drr
3-
2-
L___._
Rh,Ss 0
,
I
I
2
3
4
r
5
eq
R/a.u.
Overlap Population 0.8
0.6
1
i
0.4
i 1
T
1
total
6
I
I
7
8
(Theochem) 393 (1997) 127-139
molecular orbital diagram in which the lowest-lying valence orbital, lOa, is essentially the non-bonding N 2s orbital. This orbital is practically doubly occupied in all the states investigated. The next higher-lying orbitals are the bonding 11 u and 5a orbitals followed by the non-bonding 26 orbitals, Rh 4d6. Thereafter follows the 12a orbital which is almost non-bonding, and the 6s anti-bonding orbitals. The molecular orbital diagram described here is in agreement with our previous results regarding the carbides PdC, RhC and RuC [19]. Relative to the ‘C’ ground state, the ‘II state arises when one of the electrons occupying the 12~ orbital in the ‘C+ ground state is excited into the 6a orbitals. In the ‘C- state, the 12~ orbital is practically empty while the 67r orbitals contain approximately two electrons. The states ‘A and ‘A can both be derived from the “II state if an electron in the 26 orbitals is transferred into the 6a orbitals. Fig. 2a, Fig. 3a and Fig. 4a show the changes of the configurations of the atoms Rh and N that occur as the chemical bonds are formed in the ‘C’ ground state, Fig. 2a, and in the low-lying excited states ‘II, Fig. 3a, and ‘C-, Fig. 4a. From the overlap populations shown in Fig. 2b, Fig. 3b, and Fig. 4b, it is noted that the overlap populations as functions of the internuclear distance reach maxima approximately at the equilibrium distances for all three states. This is due to the competition between the overlap populations of the u and K orbitals. The overlap populations of the u orbitals reach maxima at approximately 4.2 a.u., i.e. considerably larger than the equilibrium internuclear distance of the molecule. At internuclear distances less than approximately 3 a.u., they even become negative. The overlap populations due to the R orbitals are increasing functions of the internuclear distance as it is reduced from 12 to 2.9 or 3.1 a.u. A similar behavior has previously been noted for the molecule MO* [21], but to us it is unexpected that the overlap populations for the ‘Cm state resemble Fig. 4. (a) Populations associated with the valence orbitals of the atoms Rh and N in the ‘Cm low-lying electronic state of RhN as derived from CASSCF wave functions. The equilibrium distance of the state is indicated by rrq. (b) Total overlap populations and also overlap populations associated with the valence o and ?r orbitals in the ‘Cm low-lying electronic state of RhN as derived from CASSCF wave functions. The equilibrium distance of the state is indicated by
R/a.u.
req.
137
Table Mulliken derived State
6 population in CASSCF Orbital
analyses
of the valence
calculations Atomic
orbltals
at the internuclear population
of the RhN distance
Overlap
molecule
in the three lowest-lying
electronic
states.
‘C’,
Orbital
analyses
‘C’
0.16
1.84
Ila
0.46
l2a l3U
0.01
0.04
0.04
0.15
1.34
0.13
0.27
0.0 I
0.24
I.81
0.15
0.02
0.60
0.02
0.08
0.05
- 0.05
0.04
0.00
I .78
0.05
I .99
I
1.41
I .93
I .20
0.1 I
0.06
I .98
0.02
0.00
0.03
0.08
_ 0.0
I .7s
1.36
0.53
0.00
0.02
1.99
0.00
I.61
3.64
0.21
0.25
- 0.06
0.00
0.00
0.18
0.00
0.22
0.40
3.97
0.00
0.00
0.00
0.00
3.91
0.00
0.00
3.97
44.31
7.02
0.67
8.87
18.03
17.75
3.94
3.40
IOU
0.16
I.85
- 0.02
0.04
0.06
0.04
1.84
0.00
I .99
II0
0.89
0.83
0.18
0.01
0.0 I
0.96
0.02
0.80
I .90
I20
0.78
0.25
0.04
0.56
0.0 I
0.2 I
0.00
0.25
I .03
I30
0.08
0.06
- 0.06
0.03
0.00
0.02
0.00
0.03
0.08 3.77
57r
2.09
1.16
0.52
0.00
0.00
2.34
0.00
I.41
6a
0.53
0.88
- 0.18
0.00
0.03
0.42
0.00
0.79
1.24
26
3.97
0.0 I
0.00
0.02
0.00
3.95
0.0 I
0.00
3.98
Total
44.38
7.08
0.54
8.64
IX.05
17.96
3.94
3.39
IO0
0.19
I .77
0.04
0.05
0.09
0.07
I .78
0.00
II0
0.86
0.85
0.17
0.01
0.03
0.9 I
0.05
0.87
I .87
I20
0.09
0.10
- 0.06
0.00
0.00
0.06
0.00
0.07
0.14
I .99
0.02
0.01
0.0 I
0.00
0.00
0.00
0.01
0.00
0.0 I
s??
2.49
0.90
0.5 1
0.00
- 0.01
2.75
0.00
1.15
3.90
6a
0.82
I .58
- 0.3 I
0.00
0.06
0.61
0.00
I .43
2.10
26
3.99
0.00
0.00
0.00
0.00
3.99
0.00
0.00
3.99
44.33
1.24
0.43
8.05
18.1 I
18.39
3.91
3.53
130
Total
P
S
6a 26
1c
-
d
P
sir
Total ‘n
N
s
IOU
‘Cm, as
number Rh
N
and
occup.
population
Rh
‘II,
of 3.3 a.u
those of the ‘C’ and “Cm states, since the state has practically no population in the 5s orbital of Rh, and therefore the 5s electrons cannot shield the 4d electrons. The observed behavior indicates that the bonding due to the c orbitals is considerably less than that due to the a orbitals. The negative contributions to the overlap populations due to the valence u orbitals is presumably due to the orthogonality constraint between the wave functions of the valence shells and the inner core shells. Fig. 2a and Fig. 3a confirm that in the case of the states ‘C+and ‘II the interaction occur between the Rh atom with the configuration (4d)‘(5s)’ and the N atom with the configuration (2s)‘(2p)‘. From Fig. 4a it is recognized that the ‘Cm state dissociates to Rh(4d)” and N(2s)‘(2p)‘. The chemical bond in the ‘C’ ground state has triple bond character due to the formation of bonding
(Tand g orbitals. The orbitals Rh 4dr and N 2px form conventional delocalized molecular orbitals. Fig. 2a and Fig. 2b indicate that the initial bonding interaction occurs between the orbitals Rh 5s and N 2~~7. As the internuclear distance is decreased, the population of the Rh 4da orbital is diminished, and thus bonding interaction can occur between the orbitals Rh 4do and N 2pa. Fig. 3a shows that the ‘II state results when Rh(4da)‘(4dx)‘(4d6)‘(5s)’ interacts with N(2pa)’ (2pa)‘. Fig. 3a and Fig. 3b indicate that the bonding interaction at large internuclear distances is mainly due to the orbitals Rh 5s and the N 2pa. At shorter internuclear distances, bonding interaction occurs between the Rh 4da and N 2~0. In Fig. 3a this change is revealed through changes in the populations of the Rh orbitals. As the internuclear distance is decreased from 8 a.u., the populations of the orbitals 4do and
4dT are diminished while the population of the 4d& orbitals is increased. The 4d6 orbitals become almost fully occupied at distances shorter than 3.6 a.u. This change of configuration enhances the bonding interaction between the orbitals N 2pa and Rh 4da, and results in the formation of a chemical bond that has triple bond character. Fig. 4a shows that the “C- state is due to the interaction between Rh(3d)“, with the configuration (4da)‘(4da)“(4dQJ and N (2pa)‘(2pa)‘. The Rh 4d6 orbitals are practically fully occupied over the entire range of internuclear distances considered. Furthermore, the 4da orbitals are fully occupied for internuclear distances larger than 5 a.u. The initial bonding interaction can occur only between the orbitals Rh 4da and N 2pa. However, at internuclear distances less than approximately 5 a.u., charge is transferred from the Rh 4dn into the N 2pa orbitals, and this allows bonding interaction between the r orbitals. The total population of the valence T orbitals amounts to approximately six electrons over the whole range of internuclear distances considered, and therefore the 6n anti-bonding orbitals have to accommodate approximately two electrons. In view Dipole moment/ Debye
2
3
4
R/L
6
7
a
Fig. 5. Dipole moments of the seven low lying electronic states, ‘E+, %I, ‘II, ‘A, ‘C-, ‘A and ‘A, of the RhN molecule as derived from CASSCF wave functions.
of this, the chemical bond in the ‘Cm state has double bond character, and it appears reasonable that the ‘Cm state has higher energy than the states ‘C’ and “II. Fig. 5 shows the dipole moments of the seven lowlying electronic states, ‘C+. “II. ‘II, ‘A, ‘Cm, ‘A, and ‘A, as derived from the CASSCF wave functions, and as functions of the internuclear distances. It is noted that the dipole moments for all the states considered are very sensitive functions of the internuclear distances. For the state ‘A the dipole moment as a function of the internuclear distance exhibits two maxima. This reflects the change of configuration that occurs for this state between 3.3 and 3.6 a.u.
7. Conclusions In the present work we have reported the results of the theoretical investigations performed for the RhN molecule. The electronic structure and the nature of the chemical bond in the RhN molecule have been elucidated through all-electron ab initio MCSCF (CASSCF) and MRCI calculations. Relativistic corrections have been included by performing perturbation calculations of the one-electron Darwin contact term and of the relativistic mass-velocity term. The inclusion of the relativistic corrections gives rise to interchanges in the low-lying states. The relativistic corrections are larger for the states with the larger populations in the Rh 5s orbital. Therefore the relativistic corrections stabilize the ‘C’ state more than the “II state, resulting in the prediction that the RhN molecule has a ‘C’ ground state. The electronic ground state of the RhN molecule, ‘C+, has the approximate orbital configuration (lOu)*( 1 lu)‘( 12u)‘(5a)‘(2&)‘. Using the results of the most elaborate calculations performed in the present work, the equilibrium distance of the RhN molecule is predicted to be 1.640 A and the vibrational frequency to be 846 cm-‘. The dipole moment has been determined to be 2.08 Debye at 3.1 a.u. On the basis of the results derived with the basis set that does not have an f function on Rh, the ‘C’ ground state is separated from the next higher-lying states, ‘II, ‘II, ‘A, “Cm, ‘A, and ‘A, by 1833,4278,6.579,8042,9632, and 13 886 cm-‘, respectively. The dissociation energy for the RhN molecule in its ‘C+ ground state is determined as 1.74 eV.
I. Shim et ul./Journal of Molecular Structure (Theochem) 393 (1997) 127-139
The chemical bond in the RhN molecule has triple bond character due to the formation of bonding K and (T molecular orbitals. The sequence of the low-lying electronic states of the RhN molecule are in accordance with a molecular orbital diagram, where the highest-lying occupied valence orbitals are the almost degenerate orbitals 6a and 120. The 26 orbitals have somewhat lower energy.
Acknowledgements The computations have been performed at the Computing Services Center at Texas A&M University, and at UNI-C at the Technical University of Denmark. IS. acknowledges the Danish Natural Research Council and the Computing Services Center at Texas A&M University for funding the computational work. The work at Texas A&M University has been supported by the Robert A. Welch Foundation and the National Science Foundation. 1.S and K.A.G. appreciate the support by NATO grant No. CRG 940581 for international collaboration in research.
References [I] I. Shim and K.A. Gingerich, Int. J. Quant. Chem., 46 (1993) 145. [Z] I. Shim and K.A. Gingerich, J. Chem. Phys., 76 (1982) 3833. [3] I. Shim and K.A. Gingerich, J. Chem. Phys., 81 (1984) 5937.
139
[4] I. Shim and K.A. Gingerich, Surface Sci., 156 (1985) 623. [S] I. Shim, H.C. Finkbeiner, and K.A. Gingerich, J. Phys. Chem., 91 (1987) 3171. [6] I. Shim, in J. Avery, J.P. Dahl and A.E. Hansen (Eds.), Understanding Molecular Properties, Reidel, Amsterdam, 1987. p. 555. [7] I. Shim, M. Pelino and K.A. Gingerich, J. Chem. Phys., 97 (I 992) 9240. [8] C.C.J. Roothaan, Rev. Mod. Phys., 32 (1960) 176. [9] J. Almlof, in Proceedings of the second seminar on computational problems in quantum chemistry, Max Planck Institut, Mtinchen, 1973. [IO] The ALCHEMY program system has been written at IBM Research Laboratory in San Jose, CA. by P.S. Bagus, B. Liu, M. Yoshimine and A.D. McLean. [II] B.O. Roos, Int. J. Quant. Chem., S14 (1980) 175. [ 121 B.O. Roos, P.R. Taylor and P.E.M. Siegbahn. Chem. Phys.. 48 (1980) 157. [13] P. Siegbahn, A. Heiberg, B. Roos and B. Levy, Phys. Ser., 21 (1980) 323. [14] P.E.M. Siegbahn, J Almlof, A. Heiberg and B.O. Roos, J. Chem. Phys., 74 (1981) 2384. [ 151 K. Andersson, M.P. Fulscher, R. Lindh, P.-A. Malmquist, J. Olsen, B.O. Roos and A.J. Sadley, University of Lund, Sweden, and P.O. Widmark, IBM Sweden, 1991. MOLCAS version 2. [16] S. Huzinaga, J. Chem. Phys., 66 (1977) 4245. [I71 S. Huzinaga, J. Chem. Phys., 54 (1971) 2283. [I 81 K.H. Huber and G. Herzberg, Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979. [ 191 K.A. Gingerich and I. Shim, in J.F.J. Todd (Ed.), Advances in Mass Spectrometry, John Wiley and Sons, New York. 1986, p. 1051. [20] C.E. Moore, Nat]. Bur. Stand Circ. No. 467, 1958, Vol. 111. [21] M.M. Goodgame and W.A. Goddard. III, Phys. Rev. Lett., 48 (1982) 135.
Journal of Molecular Structure (Theochem) 393 (I 997) 127- 139
Electronic states and nature of bonding in the molecule RhN by all-electron ab initio calculations Irene Shim”, Kim Mandixb, Karl A. Gingerich”%* “Department bDepartment
of Applied
of Applied
Chemistry,
Chemistry,
The Technical DK2800
‘Department
Univrrsir_v of Denmark,
ofElectrophysics,
DTLJ 375 and Department of Chemistry,
DTU 375, DK2800
Lyngby,
DTU 322, The Technical
Denmark
University
of Denmark,
Lyngby, Denmurk
Texas A&M
University,
Received 5 June 1996; accepted
College Station,
TX 77843, USA
25 June 1996
Abstract In the present work, all-electron ab initio multi-configuration self-consistent-field (CASSCF) and multi-reference configuration interaction (MRCI) calculations have been carried out to determine the low-lying electronic states of the molecule RhN. In addition, the relativistic corrections for the one-electron Darwin contact term and the relativistic mass-velocity correction have been determined in perturbation calculations. The spectroscopic constants for the seven low-lying electronic states have been derived by solving the SchrGdinger equation for the nuclear motion numerically. The predicted ground state of RhN is ‘C+, and this state is separated from the states 3rI, ‘II, ‘A, 3Cm,‘A and ‘A by transition energies of 1833,4278,6579,8042,9632, and I3 886 cm-‘, respectively. For the ‘C+ ground state, the equilibrium distance has been determined as 1.640 A, and the vibrational frequency as 846 cm-‘. The chemical bond in the ‘X + electronic ground state has triple bond character due to the formation of delocalized bonding a and c orbitals. The chemical bond in the RhN molecule is polar with charge transfer from Rh to N giving rise to a dipole moment of 2.08 Debye at 3.1 a.u. in the ‘C+ ground state. An approximate treatment of the spin-orbit coupling effect shows that the lowest-lying spin-orbit coupled state is O+. This state is essentially derived from the ‘2 + ground state. The second and third state, O+ and O-, mainly arise from the ‘II state. The dissociation energy of the RhN molecule in its ‘C’ ground state has been derived as I .74 eV. 0 1997 Elsevier Science B.V. Keywords:
Metal nitride; Excited state; Potential energy curve; CASSCF calculation;
1. Introduction The fixation of nitrogen in biological as well as in industrial processes requires transition metal catalysts. Therefore, basic knowledge of the chemical bonds between the nitrogen and the transition metal atoms is important for the understanding of the catalytic actions. In view of this, we have initiated
* Corresponding author.
effect
investigations of second row transition metal diatomic nitrides to elucidate their electronic structures and nature of bonding. The first molecule we have investigated is YN [ 11. Such investigations complement our previous work involving the corresponding diatomic carbides [2-71, and they will also permit comparisons of the chemical bonds and of the electronic structures of the transition metal nitrides and carbides. In the present investigation we present results of ab initio calculations for the molecule RhN. The
0166-1280/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved SO 166- 1280(96)04802-6
PII
Relativistic
128
I. Shim et al.Nournd
of Molecular
low-lying states of the RhN molecule have been studied by performing all-electron ab initio HartreeFock (HF), multi configuration self-consistent-field (CASSCF) calculations, and also multi-reference configuration interaction calculations (MRCI) that include single and double excitations from selected reference configurations. The HF calculations have been carried out in the Hartree-Fock-Roothaan formalism [8]. The integrals have been computed using the program MOLECULE [9], and for the HF calculations we have utilized the ALCHEMY program system [lo]. The multi configuration self-consistentheld calculations have been performed using the CASSCF program ([ 11-141, and the multi reference configuration interaction calculations (MRCI) have been carried out using the program system MOLCAS, version 2 [ 151. Furthermore, the relativistic effects on the low-lying electronic states of the RhN molecule have been taken into account by performing perturbation calculations of the one-electron Darwin contact term and of the relativistic mass-velocity correction. Finally, the effect of the spin-orbital coupling on the low-lying states has been considered by carrying out perturbation treatments.
2. Basis sets and HF calculations
on RhN
The basis sets consisted of contracted Gaussiantype functions. For the Rh atom, we have used a basis set that is identical to that used in our previous work on the molecule RhC [3]. It is essentially Huzinaga’s (17s 1 lp, 8d) basis [16], but it has been extended by addition of two p functions with exponents 0.20 and 0.09. These functions are needed to represent the 5p orbitals. In addition, the most diffuse s functions were contracted slightly by altering the exponents from 0.10732168 and 0.03864044 to 0.125 and 0.05, respectively. The primitive basis set (17s 13p, 8d) has been contracted to (10s 8p, 5d) using a segmented contraction scheme. In the contracted basis, the 4d orbital is represented by a triple-zeta function while all other orbitals including the unoccupied 5p orbital are represented by doublezeta functions. For the N atom we have used Huzinaga’s (10s 6p) basis [ 171, but it was augmented by a d polarization function with exponent 0.95. The basis set for the N atom has been contracted to (4s 3p, Id)
Structure (Theochrm) 393 (1997) 127-139
resulting in double-zeta representation of the s functions, triple-zeta representation of the 2p function, and a d polarization function. The above basis sets have been utilized in the major part of our calculations, but in addition we have performed further investigations using a basis set for the Rh atom that has been supplemented by an f function with exponent 1.125. In our previous work on the RhC molecule [3], we have investigated the electronic states of RhC by performing HF and valence configuration interaction (VCI) calculations. The electronic ground state and first excited state of the RhC molecule were determined to be ‘C+ and ‘II with the approximate valence shell configurations (lOc~)‘( 1 la)*( 12a)‘(57r)‘(26)’ and (lOa)*( 1la)*(5~)‘(6~)‘(26)‘, respectively. These findings are in agreement with the available spectroscopic data on the RhC molecule [ 181. Furthermore, in the HF approximation both states, ‘C+ and ?I, were found to be bound relative to the free HF atoms. In addition, it was realized that the low-lying electronic states of the RhC molecule are consistent with a simple molecular orbital diagram [6,19]. In view of the results obtained for the RhC molecule the first assumption as to the electronic ground state for the molecule RhN is ‘C+. This state arises if the additional electron in the RhN molecule, as compared to the RhC molecule, according to the build-up principle, completes the half-filled (Torbital of RhC. In Table 1 we present results of HF calculations for this ‘C+ state as well as for other states that possibly occur when the ‘S, N atom interacts with the Rh atom in either its 4F, (4d)s(5s)’ ground term or in its ID, (4d)” lowest-lying excited term. From the results reported in Table 1, it is noted that there is considerable charge transfer from the Rh to the N atom in all the states investigated. This is in accordance with the difference in electronegativities of the atoms Rh and N. More important, however, it is seen that none of the states shown are bound in the HF approximation relative to the free HF atoms. The lowest lying state identified is ‘A, and this state is unbound by 0.33 eV at the internuclear distance of 3.6 a.u. The ‘C+ state that is the first choice as a likely ground state candidate is unbound by 5.36 eV. These findings differ completely from those of the RhC molecule, where the two lowest-lying states were bound in the HF approximation. The reason for this
1. Shim et al./Journal
of Molecular
Structure
(Theochem)
393 (1997)
Table 1 Total energies for the RhN molecule as resulting from HF calculations at the internuclear moments, the gross atomic charges, and also the number of s and d electrons on Rh State
Valence shell configuration
Energy”/a.u
129
127-139
distance of 3.6 au. Also included are the dipole
Gross atomic Total number charge on Rh of d electrons
Occupation
of Rh orbitals
Dipole moment (Debye)
‘A 3c2SC+ ?rI ?I,* ‘Et % ‘A ‘A
IOU
110
120
130
5?r
6a
26
2
2 2 2 2 2 2 2 2 2
I 0 2 1 2 2 2 2 2
0 0 0 0 0 0 I 1
4 4 4 4 4 4 4 4 4
2 2 2 1
3 4 2 4 3 4 2 3 1
2 2 2 2 2 2 2
I
I 0 1 0 2
0.012148 0.043356 0.082972 0.117730 0.128343 0.196843 0.203752 0.263073 0.3 13739
0.49 0.45 0.48 0.30 0.25 0.41 0.11 0.16 0.49
18.09 18.41 17.64 18.47 18.20 18.20 17.49 17.52 16.83
4du
4dr
4ds
5s
1.15 0.48 1.69 1.03 1.69 1.85 1.80 1.88 1.85
3.93 3.93 3.95 3.43 3.52 2.35 3.79 2.64 3.98
3.00 4.00 2.00 4.00 3.00 4.00 2.00 3.00 1.00
0.32 0.06 0.78 0.16 0.47 0.28
1.09 0.96 1.27
5.48 5.1 I 5.28 3.48 3.51 5.22 1.54 1.23 2.09
’ Energy of RhN minus energy of the Rh 4F(4d)8(5s)’ and the N ‘S(2s)‘(2p)‘.
is connected to the charge transfer. In the RhC molecule the charge transferred from the Rh to the C atom could enter an empty orbital on C. In the RhN molecule, the charge transferred from Rh to N has to go into an orbital on N that is already partly occupied, and therefore the atomic correlation is larger for N as part of the RhN molecule than for the free N atom. This reveals itself as unbound states of the RhN molecule in the HF approximation. Thus, the HF approximation is not adequate for describing the electronic structure of the RhN molecule. Further investigations of the low-lying electronic states of the RhN molecule have been performed by carrying out MCSCF calculations within the framework of CASSCF.
3. Results of CASSCF calculations molecule
on the RhN
In the CASSCF calculations, the core orbitals, i.e. the Is, 2s, 3s, 4s, 2p, 3p, 4p, and 3d of Rh and the 1s orbital of N, were kept fully occupied, while the valence orbitals 5s and 4d of Rh and 2s and 2p of N, have been included in the active space. The CASSCF calculations have been performed in the subgroup CzV of the full symmetry group C,, of the RhN molecule. The calculations have been carried out for singlet, triplet, and quintet states. The number of configurations included in the CASSCF calculations
reached 1308 for the singlet states, 1746 for the triplet states, and 590 for the quintet states. The CASSCF calculations have been performed as functions of the internuclear distance, i.e. for the distances 2.9, 3.0, 3.1, 3.22, 3.28, 3.3, 3.6, 4.2, 5.0, and 12.0 a.u. The potential energies derived in the CASSCF calculations were utilized to solve the Schr iidinger equation for the nuclear motion numerically. On the basis of the results obtained, the electronic ground state of the RhN molecule is derived as ?I. This state is separated from the next higher-lying states 3Cm, ‘C+, ‘II, ‘A, ’ A, ani ‘A, by the calculated transition energies 12 11, 1475, 2130, 3809, 6479, and 7249 cm-‘, respectively. For all the states investigated, we have utilized the orbitals optimized in the CASSCF calculations at the internuclear distance of 3.3 a.u. to perform valence CI calculations. These calculations have shown that there are no additional low-lying electronic states between the states determined in the CASSCF calculations. By applying the build-up principle to the molecular orbital diagram resulting from our previous investigations of the RhC molecule, we expected the RhN molecule to have a ‘C’ electronic ground state. In view of the rather small energy separation of 1475 cm-’ between this ‘C’ state and the predicted ‘II electronic ground state, we decided to perform further investigations to ascertain the symmetry of the electronic ground state of the RhN molecule. In this connection we performed CASSCF calculations using a basis set for the Rh atom that included an f function.
I. Shim et al./Journal
130
of MolecularStructure
In addition, we also carried out more elaborate CI calculations as described in the next section. An f function with exponent 1.125 was added to the Rh basis set, and CASSCF calculations analogous to those described above were performed for the ‘C’ state at the internuclear distances 2.9, 3.0, 3.1, 3.22, 3.28, 3.3, and 3.6 a.u. Equivalent calculations were also carried out on the states “II, ‘C- and ‘II at the internuclear distances 3.0, 3.1, 3.22, 3.3, and 3.6 a.u. The potential energies derived in the CASSCF calculations were utilized to solve the Schrodinger equation for the nuclear motion numerically. The results of the calculations performed revealed that the sequence of the low-lying states does not change when an f function is added to the Rh basis set. The transition energies between the 311 state and the higher-lying ’ -, ‘E+, and ‘II, are derived as 1618, 1741, states -C and 2356 cm-’ respectively. Thus, the inclusion of the f function in the Rh basis set does not change the symmetry of the ground state, rather it enhances the separation between the lowest-lying states. Table 2 shows the spectroscopic constants obtained for the low-lying states of the RhN molecule derived on the basis of the results obtained in the CASSCF calculations, excluding and including the f function in the Rh basis set. Table 2 shows that the effect of adding the f function to the Rh basis set is reduction of the equilibrium distances and, except for the ‘.Z+ state, increase of the
(Theochem) 393 (1997) 127-139
vibrational frequencies. For the four states investigated, “II, ‘Cm, ‘C’ and ‘II, the decrease in the equilibrium distances ranges from 0.015 to 0.031 A. The vibrational frequencies increase by 67 cm-’ for the “II and ‘II states, and by 8 cm-’ for the 3Cmstate, while the vibrational frequency practically does not change for the ‘Cf state.
4. Results of MRCI calculations and ‘C’ of RhN
on the states 311
Using the Rh basis set without the f function, MRCI calculations have been performed on the states 311and ‘Cf of the RhN molecule at the internuclear distance of 3.3 a.u. In these calculations, all core orbitals have been kept frozen. Single and double excitations from the selected reference configurations have been allowed. The reference configurations were generated by keeping the 26 and the two lowest-lying valence e orbitals, 10~ and 1 lo, fully occupied while all rearrangements of six electrons within the orbitals 12a, 5?r, and 6a have been included. This resulted in 55 reference configurations for the ‘C+ state, and in 48 reference configurations for the 311state. In the final calculations, the 18 highest-lying virtual orbitals have been deleted, while all single and double excitations of all the 14 valence electrons in the reference configurations into the 36 remaining virtual
Table 2 Spectroscopic constants of the low-lying electronic states of the RhN molecule as derived from the results of CASSCF calculations sets for the Rh atom excluding and including an f function. Also shown is the population of the 5s orbital of Rh at the internuclear 3.1 a.u. State
Rh basis without f function Equilibrium distance/A
VI ?c ‘c+ ‘n TA ‘A ‘Ab ‘n
I .705 I.770 1.732 1.717 1.873 2.032 I.162 Repulsive
Vibrational frequencykm~’ 794 791 692 752 564 341 819
using basis distance of
Rh basis with an f function Transition energy/cm-’ 0 I149 1413 2360 3742 7304 9184
Dissociation energy”/eV 1.14 0.99 0.96 0.84 0.67 0.35 0.23
Population of Rh 5s at 3.1 au 0.7 1 0.05
I .oo 0.75 0.72 0.76 0.05 0.68
’ Derived as the difference between the total molecular energy at the equilibrium state. b The values shown refer to the inner minimum of the ‘A state.
Equilibrium distance/i 1.675 1.755 1.710 1.686
Vibrational frequency/cm-’
Transition energy/cm-’
861 799 691 819
0 1618 1741 2356
distance and at the internuclear
dislance of 12 a.u. for each
I. Shim et al./Journal
of Molecular Structure (Theochem) 393 (1997) 127-139
orbitals have been included. This resulted in a total of 495,727 configurations for the ‘C’ state and in 697,744 configurations for the 311 state. Relative to the CASSCF results, the gain in correlation energy is 0.198453 a.u. for the ‘C’ state, and 0.194448 a.u. for the 311 state. Although the correlation energy gained for the ‘C’ state is larger than that for the “II state, the energy of the 311state is still lower than that of the ‘C’ state, but only by 0.001976 a.u. = 434 cm-‘. In view of this small energy difference, we decided to carry out further investigations to determine the influence of the relativistic corrections, i.e. the one-electron Darwin contact term and the relativistic mass-velocity correction. In addition, we have also considered the influence of the spin-orbit coupling on the low-lying electronic states.
5. Relativistic RhN
corrections
of the low-lying
states of
Perturbation calculations have been performed to determine the corrections due to the one-electron Darwin contact term and to the relativistic mass-velocity term. The calculations have been performed using both the basis sets, i.e. the one that excludes and the one that includes the f function for Rh. The resulting perturbation energies are added to the original CASSCF energies, and the resulting potential energies
131
have been utilized to solve the SchrBdinger equation for the nuclesr motion numerically. When the relativistic corrections are taken into account, the sequence of the low-lying electronic states are interchanged, resulting in a ‘C+ electronic ground state that is separated from the higher-lying states, ‘II, ‘II, 5A, ‘C-, ‘A and ‘A, by the transition energies 1832, 4278, 6441, b036, 9635, and 13 888 cm-‘, respectively. Thus, the relativistic corrections cause an interchange of the electronic states and also an enlargement of the energy splittings between the individual states. Table 3 shows the spectroscopic constants obtained for the low-lying states of the RhN molecule derived on the basis of the results obtained in the CASSCF calculations including the relativistic corrections. Fig. 1 shows the potential energy curves for the RhN molecule as resulting from the CASSCF calculations including the relativistic corrections. By comparing Tables 2 and 3, it is noted that there is some correlation between the population of the Rh 5s orbital and the relative magnitudes of the relativistic corrections. Thus, the ‘C’ ground state that exhibits the largest relative relativistic correction has a population of l.OOe in the Rh 5s orbital at 3.1 a.u. The states “II, ‘II, ‘A have populations of 0.71, 0.75, and 0.76e, respectively, in the 5s orbital of Rh at 3.1 a.u. The relativistic corrections for these states cause similar adjustments, but less than that of the ‘C’
Table 3 Spectroscopic constants of the low-lying electronic states of the RhN molecule as derived from the results of CASSCF calculations including the relativistic corrections for the one-electron Darwin contact term and the relativistic mass-velocity correction, using bask sets for the Rh atom excluding and including an f function State
Rh basis without f function Equilibrium distance/i
‘c+ ‘n ‘n ‘A
1c
‘A ‘Ah YI
I h.57 I.676 1.698 I.795 1.757 I.946 1.751 Repulsive
Vibrational frequency/cm-’ 843 910 837 749 817 404 832
Rh basis with an f function Transition energy/cm-’ 0 1833 4278 6579 8042 9632 13886
Dissociation energy”/eV
Equilibrium distance/i
I .I4
I.640 I.657 I.674
I .5 I
Vibrational frequency/cm-’ 846 993 925
Transition energy/cm-’ 0 1752 4319
0.39 0.02
a Derived as the difference between the total molecular energy at the equilibrium state. b The values shown refer to the inner minimum of the ‘A state.
distance and at the internuclear
distance of 12 a.u. for each
132
I. Shim et al./Jourtud
of Moieculur Structure
Energy (a.u.) +4832 a.u. -0,84
-0,86
-0,88
-0,90
-0,92
-0,94
-0,96
-0,98 2
I
I
/
I
3
4
5
6
7
I
I
8
9
R (a.u.) Fig. I, Potential energy curves of 8 low-lying electronic states of the RhN molecule as derived from CASSCF calculations including the relativistic corrections for the one-electron Darwin contact term and also the correction for the relativistic mass-velocity term.
state. Therefore, the states 311and ‘C+ are interchanged resulting in the prediction that the ‘C+ state is the ground state of the RhN molecule. The states 3Cmand ‘A only have a population of 0.05e in the 5s orbital of Rh. For both these states the relativistic corrections are much smaller than for the other low-lying electronic states. Therefore these states are located at much higher energies, relative to the ground state, when the relativistic corrections are taken into account. Comparisons between Tables 2 and 3 also show that the equilibrium distances are decreased and the vibrational frequencies are increased when the relativistic corrections are taken into account. The decrease in the equilibrium distances ranges from 0.011 to 0.086 A. The vibrational frequencies are increased up to 185 cm-‘. The effect of the spin-orbit coupling on the
(Theochem) 393 (1997) 127m/39
low-lying electronic states of the RhN molecule has been investigated by carrying out an approximate treatment using the perturbation Hamiltonian H’ = Ci ~(,,)7i.~i. In addition to the diagonal spin-orbit coupling we have also considered the off-diagonal mixing between the triplet states, ‘II and 3Cm, and the singlet states, ‘II, ‘C’, and ‘A. The effect of the spin-orbit coupling is dominated by the Rh atom. Therefore, we have utilized the atomic spin-orbit coupling constant for Rh together with the atomic population on this atom as derived in the Mulliken population analyses to obtain the splittings between the molecular states. The radial integrals arising when evaluating the matrix elements of the spin-orbit coupling operator are all assumed to be equal, and the value 992 cm-’ has been obtained on the basis of the atomic spectra in Moore’s tables [ 201. The perturbation treatment has been carried out for the internuclear distances 3.0,3.1, 3.22, 3.28, 3.3, and 3.6 au. The potential energies derived have been utilized to solve the Schrodinger equation for the nuclear motion numerically. The resulting spectroscopic constants are presented in Table 4. Since the population in the open shells has only approximately 30% Rh character, the splittings between the spin-orbit coupled components are relative small. The three lowestlying states all have .I; = 0; thereafter follows a state with fl = 2, three states with n = 1, and finally a state with Q = 2. The ground state is essentially the ‘Cm state, but with some admixture of the ‘II0 state. The second and third states, both with Jz = 0, are essentially the “II,+ and ‘ITo- states. Thereafter follows a state with Q = 2. This state is mainly the ‘II2 state, but with a small admixture of the ‘A2 state. Thereafter follows three states all with Jr = 1. These states are mixtures of the states 3C-, ‘II and ‘II. Further details are presented in Table 4.
6. The low-lying molecule
electronic
states of the RhN
Table 3 shows that the energy splittings between the low-lying electronic states of RhN are not significantly influenced by the f function in the Rh basis set. Therefore the following discussion is based on the results obtained with the basis set that does not include an f function.
I. Shim et al.Nournd
of Molecular
Structure
(Theochem)
393 (1997)
133
127-139
Table 4 Spectroscopic constants of the low-lying spin-orbit coupled electronic states of the RhN molecule as derived from the results of CASSCF calculations including the relativistic corrections. i.e. the one-electron Darwin contact and the mass-velocity, using the basis set for the Rh atom that does not include an f function. Also shown are the contributions of the non-spin-orbit coupled states to each spin-orbit coupled state at the internuclear distance of 3.22 a.u. Equilibrium distance/A
State
Vibrational fequencylcm-
Transition energy/cm-’
D
Contribution ‘C’
0+
1.660 1.674 1.873 1.674 1.683 1.684 1.757
0’ O_ 2 I
I 1 2
1.75I
0
901 957 955 956 900 899 817 831
8 92 100 100 SO 50
92 8
I825 1972 2112 3074 3360 8223 14062
50 50
From Table 5, it is recognized that the major configuration in the ‘Cf ground state at internuclear distances shorter than 3.6 a.u. is the configuration (lOa)*( 1 la)‘( 12a)2(57r)4(2&)4, but with some admixture of the configuration (lOa)‘(l lt)‘(( 120)5a)~(6n)’ (26)“. Table 6 shows that this gives rise to a population of 0.40e in the 6a orbitals at 3.3 a.u. The configuration (10a)‘(lla)2(12a)2(51r)1(2~)4 is the one expected if the build-up principle is applied to the
the low-lying electronic states,
Contribution Internuclear
‘c+
‘n
3YE -
‘A
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
120
2 2 2 2 2 2 2 2 2 2 2 0 1 2 2
I 0
130
5a
2 2 2
0 0 0
I
1
2 0
0 2 I 0 0 0 0 0 0 0 0
I 1 1 1 0 2 1 0 I
I 1
1 2
‘A
100
Valence shell configuration
I10
3c
100
Table 5 The major contributions to the CASSCF wave functions describing functions of the internuclear distance
100
coupled states
‘n
TT
Table 5 shows the contributions of the major configurations to the CASSCF wave functions for four low-lying electronic states, ‘C’, ‘IT, 31T, and, ‘A, as functions of the internuclear distance. Table 6 shows the Mulliken population analyses as derived from the CASSCF wave functions for the ‘C+ electronic ground state as well as for the low-lying excited states, ‘II and ‘Cm, at the internuclear distance 3.3 a.u.
State
(‘%) of non-spin-orbit
67r 4 4 2 2 3 3
I 4 2 3 4 4 4 4 4 4 4
2.9
26 0 2 2 2 1 1 3 1 3 2 2 2 2 2 2 2 2
4 2 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3
’C ‘, ‘II, ‘Cm,and ‘A. of the RhN molecule as
of valence shell configuration
(%)
distance/au.
3.1
3.3
86
83
8
9
3.6
79 I 12
4.2
69 1 19
5.0
38 3 34
I
9 39 13 23 81 6 5
7 6 21 6 17 7 6 8 6 41 60 19 21
88 4 6
82 10 6
68 I9 13
3
89 4
1 87 5
2 83 6
90 2
88 3
89
85
I
3 77 9 2 85 3
1. Shim et al./Journal ofMolecular Structure (Theochem) 393 (1997) 127-139
134 Population
Rh,4d6
Rh.4dTi
2
---__
1
0 t 2
--m r
I
6
gq
T-~-
7
8
R/a.u.
Overlap Population
0.8;
0.6:
0.4;
molecular orbital diagram resulting from our previous investigations of the RhC molecule, i.e. if the additional electron in RhN as compared to RhC completes the half-filled 12a orbital. At the respective equilibrium distances, the low-lying excited electronic states essentially arise from the configurations (10a)2(11a)2(12a)‘(5~)3(6*)‘(26)1, the state ‘II, the states 5A and “A, and ( ~OIJ)~(1l~)2(5~)‘(67r)‘(2s)“, the states ‘C- and ‘A. In all the low-lying states, charge is transferred from the 5n to the 67r orbitals thus contributing to the radial correlation of the 7~ electrons. In the case of the %I state, the singly occupied u and ?r orbitals are triplet-coupled while they are singlet-coupled in the ‘A state. Likewise, in the states ‘C- and ‘A the two electrons in the 6a orbitals are triplet-coupled, while they are singlet-coupled in the ‘A state. The state ‘A consists of configurations in which the spins of the electrons in the 6n orbitals are singlet- as well as triplet-coupled. In accordance with Hund’s rule, the state with the highest spin angular momentum within a given configuration has the lowest energy. Table 5 shows that the state ‘A changes configuration between 3.3 and 3.6 au. The leading configuration in the ‘A state is (IOCT)~( 1 la)‘(5?r)J(6a)2(26)3 for internuclear distances of 3.3 a.u. or less. At internuclear distances 3.6 au. and larger, the configuration has changed to (lOa)‘(l Iaf2( 12a)‘(57r)‘(6?r)*(2&)‘. As seen in Fig. 1, this change of configuration reveals itself as a double minimum in the potential energy curve for the ‘A state. Fig. 2a, Fig. 3a, and Fig. 4a present the analyses of the molecular orbitals for the ‘C’ ground state and for the low-lying excited states, %I and “C-, as populations of the individual atomic orbitals and as functions of the internuclear distance. Fig. 2b, Fig. 3b and Fig. 4b show the total overlap populations as well as the overlap populations due to the valence u and r orbitals for the states, ‘C+, “II, and 3Cm,as functions of the internuclear distance.
0.2-
i 0.0
1 2
3
a
Tsq
I
4
5 R/a.u.
6
7
8
Fig. 2. a. Populations associated with the valence orbitals of the atoms Rh and N in the ‘C+ electronic ground state of RhN as derived from CASSCF wave functions. The equilibrium distance of the state is indicated by rrq. b. Total overlap populations and also overlap populations associated with the valence (r and ?r orbitals in the ‘C’ electronic ground state of RhN as derived from CASSCF wave functions. The equilibrium distance of the state is indicated by req.
1. Shim et c~l.Nournul ofMoleculur Population 47
\ ‘\
‘\
,Rh,4dd .‘-..
‘. ‘1
Rh,4dn 3i
2
1
c
,/c-.
I Rh,4du /-
,/
1 1I
0
~ 2
3T
I
A
r
4
5
eq
R/a.u.
Overlap Population 0.8
0.6
.*x
Ti
t
total
t
6
7
8
Structure (Theochrm)
135
393 (IYY7) 127-139
From Table 6, it is recognized that the I Oa orbital of all three low-lying states is essentially the non-bonding N 2s orbital. The 1 la orbital is the bonding combination of the Rh 4da and the N 2pa orbitals. In the case of the ‘C’ state, the 1 la orbital also has some admixture of the Rh 5s orbital. The 120 orbital is an essentially non-bonding orbital located mainly on the Rh atom. In the states ‘C’ and ‘II, the 12~ orbital has a large contribution from the Rh 5s orbital. The polarization of the 120 orbital results in removal of charge from the internuclear region. The 26 orbitals are the non-bonding Rh 4d6 orbitals. At 3.3 a.u., the Rh 4d6 orbitals are practically fully occupied for the three states, ‘C’, ’II, and ‘Cm, considered. The 5a and 6a orbitals are conventional bonding and anti-bonding combinations of the Rh 4d7r and the N 2p7r orbitals. Table 6 also shows that in all three states there is charge transfer from the Rh to the N atom resulting in positive gross atomic charges on Rh of 0.36e, 0.35e, and 0.46e at 3.3 a.u. in the states ‘C+, ‘7r, and ‘C-, respectively. Furthermore, it is noted that the population of the 5s orbital of Rh at 3.3 a.u. amounts to 0.87e, 0.64e, and 0.05e in the states ‘C’, ‘II, and ‘Cm, respectively. This indicates that the states ‘C+ and ‘II can both be considered as derived from the ‘F, (4d)‘(5s)’ ground term of the Rh atom, whereas the ‘C- state is derived from the ‘D, (4d)’ excited term of Rh. This finding is also consistent with the fact that coupling of the angular momenta of the ‘F, ground term of Rh with those of the ‘S, ground term of N gives rise to singlet, triplet, quintet, and septet states of C’, II, A, and @ symmetry, while the angular momenta of the ‘D, and the %i Uterms couple to triplet and quintet states of the symmetries Cm, II, and A. The results of the Mulliken population analysis indicate that the chemical bond in the ‘C+ electronic ground state of the RhN molecule has triple bond character due to the formation of the delocalized K and u orbitals. The sequence of the low-lying electronic states of the RhN molecule is consistent with a qualitative Fig. 3. Populations
associated with the valence orbitals of the atoms
Rh and N in the ‘fI
low-lymg
from CASSCF wave functions.
electronic
state of RhN as derived
The equilibrium
distance of the state
is indicated
by T_,. b. Total overlap populations
populations
associated with the valence o and ?r orbitals
low-lying
electromc
state of RhN as derived
functions.
The equilibrium
and also overlap in the ‘fl
from CASSCF
distance of the state IS indicated
wave by rc4.
1. Shim et ul.Nournal
136 Population
of MolecularStructure
Rh,4dd
t-
4-
:
Ijl
A
Rh,4drr
3-
2-
L___._
Rh,Ss 0
,
I
I
2
3
4
r
5
eq
R/a.u.
Overlap Population 0.8
0.6
1
i
0.4
i 1
T
1
total
6
I
I
7
8
(Theochem) 393 (1997) 127-139
molecular orbital diagram in which the lowest-lying valence orbital, lOa, is essentially the non-bonding N 2s orbital. This orbital is practically doubly occupied in all the states investigated. The next higher-lying orbitals are the bonding 11 u and 5a orbitals followed by the non-bonding 26 orbitals, Rh 4d6. Thereafter follows the 12a orbital which is almost non-bonding, and the 6s anti-bonding orbitals. The molecular orbital diagram described here is in agreement with our previous results regarding the carbides PdC, RhC and RuC [19]. Relative to the ‘C’ ground state, the ‘II state arises when one of the electrons occupying the 12~ orbital in the ‘C+ ground state is excited into the 6a orbitals. In the ‘C- state, the 12~ orbital is practically empty while the 67r orbitals contain approximately two electrons. The states ‘A and ‘A can both be derived from the “II state if an electron in the 26 orbitals is transferred into the 6a orbitals. Fig. 2a, Fig. 3a and Fig. 4a show the changes of the configurations of the atoms Rh and N that occur as the chemical bonds are formed in the ‘C’ ground state, Fig. 2a, and in the low-lying excited states ‘II, Fig. 3a, and ‘C-, Fig. 4a. From the overlap populations shown in Fig. 2b, Fig. 3b, and Fig. 4b, it is noted that the overlap populations as functions of the internuclear distance reach maxima approximately at the equilibrium distances for all three states. This is due to the competition between the overlap populations of the u and K orbitals. The overlap populations of the u orbitals reach maxima at approximately 4.2 a.u., i.e. considerably larger than the equilibrium internuclear distance of the molecule. At internuclear distances less than approximately 3 a.u., they even become negative. The overlap populations due to the R orbitals are increasing functions of the internuclear distance as it is reduced from 12 to 2.9 or 3.1 a.u. A similar behavior has previously been noted for the molecule MO* [21], but to us it is unexpected that the overlap populations for the ‘Cm state resemble Fig. 4. (a) Populations associated with the valence orbitals of the atoms Rh and N in the ‘Cm low-lying electronic state of RhN as derived from CASSCF wave functions. The equilibrium distance of the state is indicated by rrq. (b) Total overlap populations and also overlap populations associated with the valence o and ?r orbitals in the ‘Cm low-lying electronic state of RhN as derived from CASSCF wave functions. The equilibrium distance of the state is indicated by
R/a.u.
req.
137
Table Mulliken derived State
6 population in CASSCF Orbital
analyses
of the valence
calculations Atomic
orbltals
at the internuclear population
of the RhN distance
Overlap
molecule
in the three lowest-lying
electronic
states.
‘C’,
Orbital
analyses
‘C’
0.16
1.84
Ila
0.46
l2a l3U
0.01
0.04
0.04
0.15
1.34
0.13
0.27
0.0 I
0.24
I.81
0.15
0.02
0.60
0.02
0.08
0.05
- 0.05
0.04
0.00
I .78
0.05
I .99
I
1.41
I .93
I .20
0.1 I
0.06
I .98
0.02
0.00
0.03
0.08
_ 0.0
I .7s
1.36
0.53
0.00
0.02
1.99
0.00
I.61
3.64
0.21
0.25
- 0.06
0.00
0.00
0.18
0.00
0.22
0.40
3.97
0.00
0.00
0.00
0.00
3.91
0.00
0.00
3.97
44.31
7.02
0.67
8.87
18.03
17.75
3.94
3.40
IOU
0.16
I.85
- 0.02
0.04
0.06
0.04
1.84
0.00
I .99
II0
0.89
0.83
0.18
0.01
0.0 I
0.96
0.02
0.80
I .90
I20
0.78
0.25
0.04
0.56
0.0 I
0.2 I
0.00
0.25
I .03
I30
0.08
0.06
- 0.06
0.03
0.00
0.02
0.00
0.03
0.08 3.77
57r
2.09
1.16
0.52
0.00
0.00
2.34
0.00
I.41
6a
0.53
0.88
- 0.18
0.00
0.03
0.42
0.00
0.79
1.24
26
3.97
0.0 I
0.00
0.02
0.00
3.95
0.0 I
0.00
3.98
Total
44.38
7.08
0.54
8.64
IX.05
17.96
3.94
3.39
IO0
0.19
I .77
0.04
0.05
0.09
0.07
I .78
0.00
II0
0.86
0.85
0.17
0.01
0.03
0.9 I
0.05
0.87
I .87
I20
0.09
0.10
- 0.06
0.00
0.00
0.06
0.00
0.07
0.14
I .99
0.02
0.01
0.0 I
0.00
0.00
0.00
0.01
0.00
0.0 I
s??
2.49
0.90
0.5 1
0.00
- 0.01
2.75
0.00
1.15
3.90
6a
0.82
I .58
- 0.3 I
0.00
0.06
0.61
0.00
I .43
2.10
26
3.99
0.00
0.00
0.00
0.00
3.99
0.00
0.00
3.99
44.33
1.24
0.43
8.05
18.1 I
18.39
3.91
3.53
130
Total
P
S
6a 26
1c
-
d
P
sir
Total ‘n
N
s
IOU
‘Cm, as
number Rh
N
and
occup.
population
Rh
‘II,
of 3.3 a.u
those of the ‘C’ and “Cm states, since the state has practically no population in the 5s orbital of Rh, and therefore the 5s electrons cannot shield the 4d electrons. The observed behavior indicates that the bonding due to the c orbitals is considerably less than that due to the a orbitals. The negative contributions to the overlap populations due to the valence u orbitals is presumably due to the orthogonality constraint between the wave functions of the valence shells and the inner core shells. Fig. 2a and Fig. 3a confirm that in the case of the states ‘C+and ‘II the interaction occur between the Rh atom with the configuration (4d)‘(5s)’ and the N atom with the configuration (2s)‘(2p)‘. From Fig. 4a it is recognized that the ‘Cm state dissociates to Rh(4d)” and N(2s)‘(2p)‘. The chemical bond in the ‘C’ ground state has triple bond character due to the formation of bonding
(Tand g orbitals. The orbitals Rh 4dr and N 2px form conventional delocalized molecular orbitals. Fig. 2a and Fig. 2b indicate that the initial bonding interaction occurs between the orbitals Rh 5s and N 2~~7. As the internuclear distance is decreased, the population of the Rh 4da orbital is diminished, and thus bonding interaction can occur between the orbitals Rh 4do and N 2pa. Fig. 3a shows that the ‘II state results when Rh(4da)‘(4dx)‘(4d6)‘(5s)’ interacts with N(2pa)’ (2pa)‘. Fig. 3a and Fig. 3b indicate that the bonding interaction at large internuclear distances is mainly due to the orbitals Rh 5s and the N 2pa. At shorter internuclear distances, bonding interaction occurs between the Rh 4da and N 2~0. In Fig. 3a this change is revealed through changes in the populations of the Rh orbitals. As the internuclear distance is decreased from 8 a.u., the populations of the orbitals 4do and
4dT are diminished while the population of the 4d& orbitals is increased. The 4d6 orbitals become almost fully occupied at distances shorter than 3.6 a.u. This change of configuration enhances the bonding interaction between the orbitals N 2pa and Rh 4da, and results in the formation of a chemical bond that has triple bond character. Fig. 4a shows that the “C- state is due to the interaction between Rh(3d)“, with the configuration (4da)‘(4da)“(4dQJ and N (2pa)‘(2pa)‘. The Rh 4d6 orbitals are practically fully occupied over the entire range of internuclear distances considered. Furthermore, the 4da orbitals are fully occupied for internuclear distances larger than 5 a.u. The initial bonding interaction can occur only between the orbitals Rh 4da and N 2pa. However, at internuclear distances less than approximately 5 a.u., charge is transferred from the Rh 4dn into the N 2pa orbitals, and this allows bonding interaction between the r orbitals. The total population of the valence T orbitals amounts to approximately six electrons over the whole range of internuclear distances considered, and therefore the 6n anti-bonding orbitals have to accommodate approximately two electrons. In view Dipole moment/ Debye
2
3
4
R/L
6
7
a
Fig. 5. Dipole moments of the seven low lying electronic states, ‘E+, %I, ‘II, ‘A, ‘C-, ‘A and ‘A, of the RhN molecule as derived from CASSCF wave functions.
of this, the chemical bond in the ‘Cm state has double bond character, and it appears reasonable that the ‘Cm state has higher energy than the states ‘C’ and “II. Fig. 5 shows the dipole moments of the seven lowlying electronic states, ‘C+. “II. ‘II, ‘A, ‘Cm, ‘A, and ‘A, as derived from the CASSCF wave functions, and as functions of the internuclear distances. It is noted that the dipole moments for all the states considered are very sensitive functions of the internuclear distances. For the state ‘A the dipole moment as a function of the internuclear distance exhibits two maxima. This reflects the change of configuration that occurs for this state between 3.3 and 3.6 a.u.
7. Conclusions In the present work we have reported the results of the theoretical investigations performed for the RhN molecule. The electronic structure and the nature of the chemical bond in the RhN molecule have been elucidated through all-electron ab initio MCSCF (CASSCF) and MRCI calculations. Relativistic corrections have been included by performing perturbation calculations of the one-electron Darwin contact term and of the relativistic mass-velocity term. The inclusion of the relativistic corrections gives rise to interchanges in the low-lying states. The relativistic corrections are larger for the states with the larger populations in the Rh 5s orbital. Therefore the relativistic corrections stabilize the ‘C’ state more than the “II state, resulting in the prediction that the RhN molecule has a ‘C’ ground state. The electronic ground state of the RhN molecule, ‘C+, has the approximate orbital configuration (lOu)*( 1 lu)‘( 12u)‘(5a)‘(2&)‘. Using the results of the most elaborate calculations performed in the present work, the equilibrium distance of the RhN molecule is predicted to be 1.640 A and the vibrational frequency to be 846 cm-‘. The dipole moment has been determined to be 2.08 Debye at 3.1 a.u. On the basis of the results derived with the basis set that does not have an f function on Rh, the ‘C’ ground state is separated from the next higher-lying states, ‘II, ‘II, ‘A, “Cm, ‘A, and ‘A, by 1833,4278,6.579,8042,9632, and 13 886 cm-‘, respectively. The dissociation energy for the RhN molecule in its ‘C+ ground state is determined as 1.74 eV.
I. Shim et ul./Journal of Molecular Structure (Theochem) 393 (1997) 127-139
The chemical bond in the RhN molecule has triple bond character due to the formation of bonding K and (T molecular orbitals. The sequence of the low-lying electronic states of the RhN molecule are in accordance with a molecular orbital diagram, where the highest-lying occupied valence orbitals are the almost degenerate orbitals 6a and 120. The 26 orbitals have somewhat lower energy.
Acknowledgements The computations have been performed at the Computing Services Center at Texas A&M University, and at UNI-C at the Technical University of Denmark. IS. acknowledges the Danish Natural Research Council and the Computing Services Center at Texas A&M University for funding the computational work. The work at Texas A&M University has been supported by the Robert A. Welch Foundation and the National Science Foundation. 1.S and K.A.G. appreciate the support by NATO grant No. CRG 940581 for international collaboration in research.
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