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Theoretical study of bending and symmetric stretching vibrational levels of the lowest five quintet and two triplet states of FeH2
Chemical Physics Letters 388 (2004) 389–394 www.elsevier.com/locate/cplett
Theoretical study of bending and symmetric stretching vibrational levels of the lowest five quintet and two triplet states of FeH2 Kiyoshi Tanaka *, Katsuyuki Nobusada Division of Chemistry, Graduate School of Science, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo 060-0810, Japan Received 11 November 2003; in final form 23 February 2004 Published online:
Abstract Adiabatic potential surfaces of the lowest two triplet states and the lowest five quintet states of FeH2 were calculated using ab initio multi-reference singly and doubly excited configuration interaction plus Davidson’s type correction. Multi-reference coupled pair approximation was applied to obtain more accurate term energies of the triplet states. In contrast to the quintet states, which are known to be linear, the lowest two triplet states are strongly bent. Vibrational analyses of both quintet and triplet states were carried out. We discuss the effect of non-adiabatic coupling on the bending vibrational levels of 5 B2 and (2)5 A1 through a conical intersection. Ó 2004 Elsevier B.V. All right reserved.
1. Introduction FeH2 was identified in matrix isolation studies [1,2] in the middle of 1980s. At the almost same time, theoretical studies were carried out by Siegbahn et al. [3] and Granucci and Persico [4]. The two papers predicted a linear 5 Dg electronic ground state. Recently Marian and co-workers reported theoretical two studies of finestructure effect in the anti-symmetric stretching vibrational spectrum [5], and in the bending and symmetric stretching vibronic levels [6] of the quintet states of FeH2 and FeD2 . The first detection of FeH2 in the gas phase was carried out by Brown’s group [7] in 1996, by the technique of laser magnetic resonance. Their analysis indicates that the ground state is in the 5 Dg state in agreement with the previous calculations. They further published two papers concerning experimental analyses of free FeD2 and FeH2 [8,9]. In contrast to the quintet states, the lower triplet states of FeH2 have been less studied. As was discussed in the previous works [3,4], it would be helpful to study the lower triplet states of FeH2 theoretically by accurate *
Corresponding author. Fax: +81-11-706-4921. E-mail address: [email protected] (K. Tanaka).
0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All right reserved. doi:10.1016/j.cplett.2004.03.032
calculations of their energetic position and their vibration levels. The lower triplet states are expected to be strongly bent as was reported by the preliminary calculations [3,4]. The electronic structure of the lowest two quartet states of FeH is also indicative of the bent structure of the triplet states. The lowest two triplet states correlate to the addition of H to the lowest two quartet states of FeH. The open shells of the main configuration of the lowest two quartet states of FeH are attributed to 3dp and 3dd orbitals [10]. The r-unpaired electron directed toward the outside of the bond is not included. This implies that the triplet states of FeH2 prefer to have bent structure because linear r bond is not expectable. Very extensive studies on vibrational levels of the quintet states were conducted by Marian’s group [6] including relativistic effects. They, however, did not consider a non-adiabatic effect caused by potential surface crossing between the 5 B2 and (2)5 A1 states against change in the bond angle. These two states both belong to the 5 A0 states when the molecule distorts into the Cs symmetry. This means that the surfaces include a conical intersection between the two states. In such a case, anti-symmetric stretching vibration induces non-adiabatic coupling between the bending levels of the two states, as is discussed on NO2 [11,12]. In this
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work, bending and symmetric stretching vibrational levels will be solved firstly using the adiabatic potential surfaces of the quintet and triplet states. We will discuss how the non-adiabatic coupling affects the vibrational levels. In the next section, method of calculation will be given. The results and discussion will be presented in Section 3.
2. Method of calculation The molecular point group of C2v was employed in the calculation. The molecule was placed on the yz plane where the molecular axis was taken to be along the z-axis. We used extensive contracted gaussian type orbitals (CGTOs) for Fe [8s6p5d2f1g]/(22s16p11d4f2g), and for H [4s3p]/(6s3p), which were used in our calculation on FeH [10]. Multi-configuration self-consistent field (MCSCF) calculations were carried out. In the MCSCF calculations, 10 valence electrons were distributed among the following nine molecular orbitals (MO) {6a1 , 7a1 , 8a1 , 3b1 , 4b1 , 3b2 , 4b2 , 1a2 , and 2a2 }, and single and double excitations to a MO set {9a1 , 10a1 , 11a1 , 5b2 , and 6b2 } were included. Those MO’s of the former set are composed of 3d and Fe–H bonding which are occupied in a leading configuration and Fe–H antibonding and d describing left-right correlation in the Fe–H bond and in–out correlation in the 3d shell, respectively. The MO’s of the latter set are the Fe–H antibonding and d in the a1 and b2 manifolds. Two states averaged MCSCF calculations were applied to the 5 A1 symmetry species, because one 5 component of 5 Dg and 5 Rþ g belongs to the A1 symmetry species in the bent structure. After MCSCF calculations, configuration interaction (CI) calculations were carried out, where we employed the selected reference configuration state functions (CSF’s) plus singly and doubly excited CSF’s relative to the reference function(s). The reference CSF’s were selected using the results of the MCSCF wavefunctions. The numbers of the reference CSF’s were 4, 2, 1, 1, 7, and 9 for 5 A1 , 5 B1 , 5 B2 , 5 A2 , 3 B1 , and 3 A2 , respectively. The number of CSF’s were 394,149, 248,159, 132,573, 126,230, 964,161, and 762,139 for 5 A1 , 5 B1 , 5 B2 , 5 A2 , 3 B1 and 3 A2 , respectively. The Davidson type quadruple correction [13,14] was taken into account after the CI calculations (MRSDCI + Q). These calculations were carried out using Alchemy II [15–17]. The multi-reference coupled pair approximation (MRCPA) [18,19] was also applied to the ground state and the triplet states to obtain accurate term energies. Because the computational scheme for the 5 A1 states was different from other states, the potential curves at the linear structure of the two components of the 5 Dg
(X5 A1 and 5 B1 ) states did not coincide with each other. The surfaces of the 5 B1 , 5 B2 , 5 A2 , 3 B1 and 3 A2 states were shifted down by 0.0305 eV (246 cm1 ) to make the bottom of the potential surface of the X5 A1 state degenerate with that of the 5 B1 state at the linear structure. Vibrational levels of the symmetric stretching and bending modes were calculated using a program code developed by one of the authors (Nobusada [20]). For the sake of considering the non-adiabatic effects caused by the conical intersection between the 5 B2 and (2)5 A1 states, we developed a set of coupled equations of the bending vibration and solved them. The method will be described in Appendix A.
3. Results and discussion 3.1. Equilibrium structures The quintet states were stabilized at symmetric and linear structure in accordance with the previous results. The bonding scheme of the primary electronic configuration of the quintet states was qualitatively expressed in the following form: 2
2
3d6 ð4sFe 1sH : rg Þ ð4pFe 1sH : ru Þ :
ð1Þ 3
On the other hand, the lowest two triplet states, B1 and A2 , were strongly bent. Their equilibrium bond angles, (he ), were both 101° as shown in Table 1. Both states were approximately represented by the following configuration in which 3d orbital participates in the FeH bond. 3
2
2
3d6 ð4s4pFe 1sH : a1 Þ ð3dFe 1sH : b2 Þ :
ð2Þ
Table 1 Equilibrium bond angle ðhe Þ of the lowest two triplet states and bond distances ðRe Þ of the triplet and quintet states Theory
he (°) 3 B1 3 A2
SBBb
MMPc
KUBd
KMLUTBe
101 101
– –
– –
– –
– –
1.688 1.672 – – –
1.653 1.635 1.634 – –
1.648 – – – –
1.665 – – – –
f
Re (A) 5 Dg 5 Pg 5 þ Rg 3 B1 3 A2 a
Experiment
Presenta
1.665 1.649 1.651 1.540 1.540
Results by MRSDCI + Q. Ref. [3]. c Ref. [6]. d Ref. [9]. e Ref. [7]. f Linear structure of the quintet states is established. b
K. Tanaka, K. Nobusada / Chemical Physics Letters 388 (2004) 389–394
Table 1 exhibits the calculated equilibrium bond distances, (Re ). The previous results are also included in the table. The ground state equilibrium distance of is in good agreement with the value observed 1.665 A by K€ orsgen and co-workers [7]. The equilibrium bond distances of the excited states, 5 Pg and 5 Rþ g , have not been observed. Judging from the level of approximation employed in the present work, the bond distances of 5 Pg and 5 Rþ g are expected to be close to those predicted values. The equilibrium bond distances of the triplet states are shorter than those of the quintet states, reflecting participation of 3d orbital in the FeH bond. 3.2. Term energies and lower vibration levels The term energy, Te (5 Pg 5 Dg ), obtained by the present calculation is 1861 cm1 and close to Marian’s value of 1783 cm1 (see Table 2). These values are close to the observed excitation energy of the anti-symmetric vibration, 1674 cm1 , of the ground state [9]. Since the excitation intensity of the anti-symmetric vibration should be stronger than that of the forbidden electronic excitation, 5 Pg 5 Dg , it is inferred that this situation makes it difficult to observe the T0 value for the latter transition.
Table 2 Term energies, Te (cm1 ), of the lowest two triplet states and the lowest two quintet states Theory
5
Pg 5 þ Rg 3 B1 3 A2 3
391
The term energies of the triplet states, Te (3 B1 X5 Dg ) and Te (3 A2 X5 Dg ), obtained by MRSDCI + Q were 1.16 104 cm1 (1.44 eV) and 1.22 104 cm1 (1.51 eV), respectively. It is noted that the 3 B1 , 3 A2 and 5 Dg states of FeH2 correlate to the lowest 4 D, 4 P and 6 D of FeH, among which the electron correlation in 4 D and 4 P is stronger than in 6 D [10]. The MRCPA method was applied to the 3 B1 , 3 A2 states and the ground state at their equilibrium geometries. As shown in Table 2, the predicted term energies, Te (3 B1 X5 Dg ) and Te (3 A2 X5 Dg ), were 1.00 104 cm1 (1.24 eV) and 1.06 104 cm1 (1.32 eV), respectively. They are about 0.2 eV lower than the MRSDCI + Q results. Because better agreement with experimental values is provided by MRCPA in FeH, we think these values are more reliable than those by MRSDCI + Q. Lower vibrational levels were calculated using the present adiabatic potential surfaces. Table 3 exhibits lower seven energy levels of the 5 Dg (X5 A1 , 5 B1 ), 5 5 Pg (5 A2 , 5 B2 ) and 5 Rþ g ((2) A1 ) states. The lowest four vibrational levels of the triplet states of FeH2 are shown in Table 4. The levels are designated by (mss ; mb ), where mss and mb are quantum number of the symmetric stretching and bending vibration modes, respectively. The (0,1)–(0,0) energy difference of the 5 Dg state of FeD2 , 272 or 269 cm1 , is close to Marian’s value of 250 cm1 [6] and they are comparable to the gas phase
Table 4 Lower bending vibrational levelsa and harmonic frequency (xss ) of the triplet states of FeH2 (in cm1 )
Presenta
SBBb
GPc
MMPd
(mss ; mb )b
3
1861 2695 1.00 104 1.06 104
2016 – – –
2134 2273 1.5 104 –
1783 2067 – –
(0,0) (0,1)–(0,0) (0,2)–(0,0) (0,3)–(0,0)
1063 328 693 1071
1162 464 927 1387
xss c
1798
1860
a 3 Results by MRSDCI + Q for 5 Pg , 5 Rþ g and by MRCPA(4) for B1 , A2 . b Ref. [3]. c Ref. [4]. d Ref. [6].
3
B1
A2
a
Same as (a) in Table 3. b Same as (b) in Table 3. c xss ¼ 3 ð0; 0Þ ð0; 1Þ.
Table 3 Lower vibration levelsa of the quintet states of FeH2 and FeD2 (in cm1 ) (mss ; mb )b
(0,0) (0,1)–(0,0) (0,2)–(0,0) (0,3)–(0,0) (0,4)–(0,0) (0,5)–(0,0) (1,0)–(0,0) a b
FeH2
FeD2
X5 A1
5
1078 379 765 1158 1556 1962 1821
1079 375 758 1147 1542 1943 1825
B1
5
A2
1061 343 700 1068 1445 1843 1834
B2
(2)5 A1
X5 A1
5
1205 620 1233 1839 2438 3031 1839
1029 291 613 956 1317 1697 1846
758 272 547 827 1109 1393 1247
759 269 542 819 1099 1381 1250
5
They are relative to the bottom of the potential surface of each state. mss : symmetric stretching, mb : bending.
B1
5
A2
739 245 498 756 1020 1289 1256
B2
(2)5 A1
845 450 896 1338 1776 2211 1253
714 203 424 659 904 1158 1259
5
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K. Tanaka, K. Nobusada / Chemical Physics Letters 388 (2004) 389–394
observed values 226 and 221 cm1 [8]. Those of FeH2 , 379 and 375 cm1 , are also close to Marian’s value of 354 cm1 [6]. Since unharmonicity is expected to be small, we calculated the bending energies, hxb , by using the (0,1)–(0,0) difference of the 3 B1 and 3 A2 states. They are 328 and 464 cm1 , respectively and almost the same order of those of the quintet states. The symmetric stretching vibration frequency, xss , was estimated assuming that the zero point energy of (0,0) is equal to hðxss þ xb Þ=2 and using hxb given by the (0,1)–(0,0) difference. The estimated values of the quintet states lie in the range of 1779 12 cm1 and those of 3 B1 and 3 A2 are 1798 and 1860 cm1 , respectively. 3.3. Non-adiabatic effects on the bending vibrational levels of the 5 B2 and (2)5 A1 states In the previous subsection, we neglected the nonadiabatic effect caused by a crossing between 5 B2 and (2)5 A1 (see Fig. 1). They both belong to the 5 A0 symmetry species when the molecule is distorted into the Cs point group. Owing to the crossing, the bending energy levels of these two states couple with each other through the anti-symmetric stretching vibration. The lowest six bending vibrational levels are presented in the figure, where the zero point energies are estimated by the half of the difference between (0,1) and (0,0) levels. Since these levels are close to the crossing point, it is desirable
Fig. 1. The bending potential curves of the 5 B2 (solid curve) and (2)5 A1 (broken curve) states and the lower bending vibrational levels: broken lines; the lowest three vibrational levels of 5 B2 and dotted lines; the lowest three vibrational level of (2)5 A1 .
to analyze the effect of non-adiabatic coupling. We derived a set of coupled equations of bending vibration, assuming the followings: (1) We assume that the symmetric and anti-symmetric stretching modes are not excited, because we are interested in the lower bending states and the frequencies of the both stretching modes are much larger than the bending frequencies. (2) Since the potential surfaces show that the variables of the symmetric stretching (FeH distance) and the bending (HFeH angle) are separable, the FeH distance of the symmetric motion, R, is kept constant (3.118 a.u. (1.650 A)). (3) We assume that the variable of the anti-symmetric mode is also separable from the other two modes. In the following discussion, we employ diabatic basis originating from 5 B2 and 5 A1 whose wavefunctions are expressed as wB2 and wA1 , respectively. Let us express wavefunctions of anti-symmetric stretching by /Bi 2 and 1 1 /A and bending wavefunctions by vBvb2 and vA j v0b . The 0 suffices i; j; vb ; and vb are vibrational quantum number. We consider the following two cases; wB2 /B0 2 vBvb2 perA1 A1 1 A1 turbed by wA1 /A 1 vv0b , and wA1 /0 vvb perturbed by B2 B2 wB2 /1 vv0 . When the anti-symmetric vibrational varib able is defined as r ¼ ðRFeH1 RFeH2 Þ=2 and the indices X and Y are designated as B2 or A1 , the coupling term for the bending motion is h/X1 jHXY j/Y0 i and further Y o/ j Y h/Xi jwX ow or j or i will contribute to the coupling. The Hamiltonian matrix element, HXY , is not zero when the molecule is distorted into the Cs symmetry. In this work, we consider only the former type of coupling which is discussed well [12] in NO2 . The equations and approximations are described in Appendix A. Table 5 shows the lower eight energy levels of FeH2 (the column represented by non-adiabatic). In addition to the column, the results without coupling are given in the column represented by diabatic. The two dimensional diabatic results (uncoupled between the two electronic states) are included to demonstrate how the variable separation is a reasonable approximation. The good agreement of one dimensional uncoupled results with the two dimensional ones indicates separability of the variables of bending and symmetric stretching. It is found that the non-adiabatic levels are shifted down by about 8–17 cm1 in comparison with the diabatic results. The shifts are independent of the relative separation of the levels. Table 6 shows overlap matrix elements between coupled wavefunctions and uncoupled (diabatic) wavefunctions. The wavefunctions are represented almost by a single wavefunction, although the mixing weights are gradually increasing in the higher levels. Very small energy shifts and weak mixing among the bending levels indicate that the nuclear bending motion of the two states is almost diabatic, as far as the lower levels concern.
K. Tanaka, K. Nobusada / Chemical Physics Letters 388 (2004) 389–394
393
Table 5 Non-adiabatic effect on the lower bending vibration through the conical intersection between 5 B2 and (2)5 A1 (in cm1 )a of FeH2 Coupled state number
Leading state
First Second Third Fourth Fifth Sixth Seventh Eighth
5
Two dimensionalb
Eqs. (A.1) and (A.2)
B2 B2 (2)5 A1 (2)5 A1 5 B2 (2)5 A1 (2)5 A1 5 B2 5
mb c
Non-adiabatic
Diabatic
Diabatic
0 1 0 1 2 2 3 3
303 922 954 1258 1536 1585 1931 2142
311 931 971 1272 1547 1600 1949 2159
310 930 974 1265 1543 1587 1931 2149
a
Relative to the bottom of the potential surfece of 5 B2 . From the results given by symmeric stretching and bending vibrations. c Bending quantum number. b
Table 6 Overlap between the coupled wavefunctions and diabatic wavefunctions for 5 B2 and (2)5 A1 of FeH2 Electronic state
5
mb a
B2
Coupled wavefunction First
Second
Third
Fourth
b
0 0.9984 0.0 )0.0005 0.0 )0.0001
0 0.0 0.9979 0.0 )0.0014 0.0
1 )0.1218 0.0 0.0157 0.0 0.0045
1 0.0 0.0922 0.0 )0.0246 0.0
ma b
1 0.0547 0.0 )0.0102 0.0 )0.0024
1 0.0 )0.0620 0.0 0.0160 0.0
0 0.9924 0.0 0.0019 0.0 0.0008
0 0.0 0.9954 0.0 )0.0008 0.0
ma 0 1 2 3 4
(2)5 A1 0 1 2 3 4 a b
Fifth
Sixth
Seventh
Eighth
0 0.0014 0.0 0.9967 0.0 0.0033
1 )0.0685 0.0 )0.0747 0.0 )0.0325
1 0.0 0.0495 0.0 )0.0360 0.0
0 0.0 0.0047 0.0 0.9933 0.0
1 )0.0319 0.0 ).0707 0.0 )0.0210
0 )0.0090 0.0 0.9941 0.0 )0.0039
0 0.0 )0.0043 0.0 0.9949 0.0
1 0.0 0.0766 0.0 0.0821 0.0
Bending quantum number. Antisymmetric stretching quantum number.
4. Conclusions The accurate adiabatic potential surfaces of the lowest two triplet states of FeH2 have been obtained by the MRSDCI + Q level of theory as well as the lowest five quintet states. The MRCPA method is also used to obtain accurate term energies of the triplet states. The lower symmetric stretching and bending vibration levels are calculated. The lowest two triplet states, 3 B1 and 3 A2 , are strongly bent in contrast to the lowest five quintet states. The electronic ground state is 5 Dg in agreement with the previous theoretical and experimental results. The triplet states lie above the ground state by 1.2–1.3 eV, whereas the quintet excited states are within 0.3 eV from the ground state. We have studied non-adiabatic coupling between the bending modes of the 5 B2 and (2)5 A1 states whose surfaces include a conical intersection. As far as the lower levels concern, the energy shifts caused by the coupling are very small and the nuclear motion of the two states
is almost diabatic. The shifts are independent of the relative separation between vibrational levels.
Acknowledgements This work was partially supported by a Grant-InAid for Scientific Research on a Priority Area (A), No. 12042201, from the Japanese Ministry of Education, Science, Sports, and Culture. We are grateful to a referee’s kind comments about non-adiabatic coupling.
Appendix A We derived the following set of coupled equations with the assumption shown in Section 3.3. The suffices 1 and 2 in the following two equations represent the 5 B2 or (2)5 A1 state. The suffix 1 indicates primary state and 2 means perturbing state.
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K. Tanaka, K. Nobusada / Chemical Physics Letters 388 (2004) 389–394
h2 d 2 þ W1 ðhÞ v1 ðhÞ þ V12 ðhÞv2 ðhÞ ¼ Ev1 ðhÞ; 2lR2 dh2 ðA:1Þ
2
2
the frequencies for the two electronic states. The value is nearly equal to that of the ground state. Fitting of V12 is carried out in the form V12 ¼ a þ bh2 þ ch4 :
h d þ hxa þ W2 ðhÞ v2 ðhÞ þ V21 ðhÞv1 ðhÞ ¼ Ev2 ðhÞ; 2lR2 dh2 ðA:2Þ
where h is defined as h ¼ ðp \HFeHÞ, W1 and W2 are adiabatic potential surfaces of the two states, and v1 and v2 are bending vibrational wavefunction of the two states. The reduced mass l is given by 2MH MFe ; ðA:3Þ 2MH þ MFe where MH and MFe are the mass of hydrogen and iron. The coupling term, V12 ð¼ V21 Þ, is defined as follows:
ðA:7Þ
a, b, and c are 0.000609, )0.000130, and 0.00103 in atomic unit for FeH2 . The surface crossing between the two states takes place at 10° of h by MRSDCI + Q calculation and near 27.4° by the MCSCF calculation. In applying the Eq. (A.7), we scaled the h by a factor 2.74, i.e. 2:74h. In solving Eqs. (A.1) and (A.2), we used a discrete variable representation [21].
l¼
ð1Þ ð2Þ V12 ¼ h/0 jH12 j/1 i; ð1Þ ð2Þ where /0 and /1 are
ðA:4Þ
the vibrational wavefunctions of the anti-symmetric stretching mode of the lowest level of the primary state and first excited level of the perturving state, respectively. H12 is a matrix element of the electronic Hamiltonian between the two electronic wavefunctions. Evaluation of H12 was carried out using the results of three states averaged MCSCF calculations for the 5 A0 states in Cs (X5 A1 , 5 B2 and (2)5 A1 in C2v ). The calculation was carried out at deformation variable, r, of 0.0, 0.025, 0.050, 0.075, 0.1, 0.15, 0.2 and 0.25 a.u. where R ¼ ðRFeH1 þ RFeH2 Þ=2 are kept to be 3.118 a.u. H12 was obtained from the following equation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA:5Þ DE ¼ DE02 þ 4H212 ; where DE and DE0 are energy separations between the second and third 5 A0 states at r and r ¼ 0, respectively. The matrix element H12 is fitted in the following form at each h (0°, 5°, 10°, 15°, 20°, 22°, 24°, 26°, 27°, 27.3°, 27.5°, 28°, 29°, 30°, 32°, 33°, and 40°) H12 ¼ ar þ br3 ;
ðA:6Þ
where a and b are determined by a least square technique. In evaluating V12 , we assumed harmonic vibrað1Þ ð2Þ tional wavefunctions for /0 and /1 with 1700 cm1 of
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Theoretical study of bending and symmetric stretching vibrational levels of the lowest five quintet and two triplet states of FeH2 Kiyoshi Tanaka *, Katsuyuki Nobusada Division of Chemistry, Graduate School of Science, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo 060-0810, Japan Received 11 November 2003; in final form 23 February 2004 Published online:
Abstract Adiabatic potential surfaces of the lowest two triplet states and the lowest five quintet states of FeH2 were calculated using ab initio multi-reference singly and doubly excited configuration interaction plus Davidson’s type correction. Multi-reference coupled pair approximation was applied to obtain more accurate term energies of the triplet states. In contrast to the quintet states, which are known to be linear, the lowest two triplet states are strongly bent. Vibrational analyses of both quintet and triplet states were carried out. We discuss the effect of non-adiabatic coupling on the bending vibrational levels of 5 B2 and (2)5 A1 through a conical intersection. Ó 2004 Elsevier B.V. All right reserved.
1. Introduction FeH2 was identified in matrix isolation studies [1,2] in the middle of 1980s. At the almost same time, theoretical studies were carried out by Siegbahn et al. [3] and Granucci and Persico [4]. The two papers predicted a linear 5 Dg electronic ground state. Recently Marian and co-workers reported theoretical two studies of finestructure effect in the anti-symmetric stretching vibrational spectrum [5], and in the bending and symmetric stretching vibronic levels [6] of the quintet states of FeH2 and FeD2 . The first detection of FeH2 in the gas phase was carried out by Brown’s group [7] in 1996, by the technique of laser magnetic resonance. Their analysis indicates that the ground state is in the 5 Dg state in agreement with the previous calculations. They further published two papers concerning experimental analyses of free FeD2 and FeH2 [8,9]. In contrast to the quintet states, the lower triplet states of FeH2 have been less studied. As was discussed in the previous works [3,4], it would be helpful to study the lower triplet states of FeH2 theoretically by accurate *
Corresponding author. Fax: +81-11-706-4921. E-mail address: [email protected] (K. Tanaka).
0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All right reserved. doi:10.1016/j.cplett.2004.03.032
calculations of their energetic position and their vibration levels. The lower triplet states are expected to be strongly bent as was reported by the preliminary calculations [3,4]. The electronic structure of the lowest two quartet states of FeH is also indicative of the bent structure of the triplet states. The lowest two triplet states correlate to the addition of H to the lowest two quartet states of FeH. The open shells of the main configuration of the lowest two quartet states of FeH are attributed to 3dp and 3dd orbitals [10]. The r-unpaired electron directed toward the outside of the bond is not included. This implies that the triplet states of FeH2 prefer to have bent structure because linear r bond is not expectable. Very extensive studies on vibrational levels of the quintet states were conducted by Marian’s group [6] including relativistic effects. They, however, did not consider a non-adiabatic effect caused by potential surface crossing between the 5 B2 and (2)5 A1 states against change in the bond angle. These two states both belong to the 5 A0 states when the molecule distorts into the Cs symmetry. This means that the surfaces include a conical intersection between the two states. In such a case, anti-symmetric stretching vibration induces non-adiabatic coupling between the bending levels of the two states, as is discussed on NO2 [11,12]. In this
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work, bending and symmetric stretching vibrational levels will be solved firstly using the adiabatic potential surfaces of the quintet and triplet states. We will discuss how the non-adiabatic coupling affects the vibrational levels. In the next section, method of calculation will be given. The results and discussion will be presented in Section 3.
2. Method of calculation The molecular point group of C2v was employed in the calculation. The molecule was placed on the yz plane where the molecular axis was taken to be along the z-axis. We used extensive contracted gaussian type orbitals (CGTOs) for Fe [8s6p5d2f1g]/(22s16p11d4f2g), and for H [4s3p]/(6s3p), which were used in our calculation on FeH [10]. Multi-configuration self-consistent field (MCSCF) calculations were carried out. In the MCSCF calculations, 10 valence electrons were distributed among the following nine molecular orbitals (MO) {6a1 , 7a1 , 8a1 , 3b1 , 4b1 , 3b2 , 4b2 , 1a2 , and 2a2 }, and single and double excitations to a MO set {9a1 , 10a1 , 11a1 , 5b2 , and 6b2 } were included. Those MO’s of the former set are composed of 3d and Fe–H bonding which are occupied in a leading configuration and Fe–H antibonding and d describing left-right correlation in the Fe–H bond and in–out correlation in the 3d shell, respectively. The MO’s of the latter set are the Fe–H antibonding and d in the a1 and b2 manifolds. Two states averaged MCSCF calculations were applied to the 5 A1 symmetry species, because one 5 component of 5 Dg and 5 Rþ g belongs to the A1 symmetry species in the bent structure. After MCSCF calculations, configuration interaction (CI) calculations were carried out, where we employed the selected reference configuration state functions (CSF’s) plus singly and doubly excited CSF’s relative to the reference function(s). The reference CSF’s were selected using the results of the MCSCF wavefunctions. The numbers of the reference CSF’s were 4, 2, 1, 1, 7, and 9 for 5 A1 , 5 B1 , 5 B2 , 5 A2 , 3 B1 , and 3 A2 , respectively. The number of CSF’s were 394,149, 248,159, 132,573, 126,230, 964,161, and 762,139 for 5 A1 , 5 B1 , 5 B2 , 5 A2 , 3 B1 and 3 A2 , respectively. The Davidson type quadruple correction [13,14] was taken into account after the CI calculations (MRSDCI + Q). These calculations were carried out using Alchemy II [15–17]. The multi-reference coupled pair approximation (MRCPA) [18,19] was also applied to the ground state and the triplet states to obtain accurate term energies. Because the computational scheme for the 5 A1 states was different from other states, the potential curves at the linear structure of the two components of the 5 Dg
(X5 A1 and 5 B1 ) states did not coincide with each other. The surfaces of the 5 B1 , 5 B2 , 5 A2 , 3 B1 and 3 A2 states were shifted down by 0.0305 eV (246 cm1 ) to make the bottom of the potential surface of the X5 A1 state degenerate with that of the 5 B1 state at the linear structure. Vibrational levels of the symmetric stretching and bending modes were calculated using a program code developed by one of the authors (Nobusada [20]). For the sake of considering the non-adiabatic effects caused by the conical intersection between the 5 B2 and (2)5 A1 states, we developed a set of coupled equations of the bending vibration and solved them. The method will be described in Appendix A.
3. Results and discussion 3.1. Equilibrium structures The quintet states were stabilized at symmetric and linear structure in accordance with the previous results. The bonding scheme of the primary electronic configuration of the quintet states was qualitatively expressed in the following form: 2
2
3d6 ð4sFe 1sH : rg Þ ð4pFe 1sH : ru Þ :
ð1Þ 3
On the other hand, the lowest two triplet states, B1 and A2 , were strongly bent. Their equilibrium bond angles, (he ), were both 101° as shown in Table 1. Both states were approximately represented by the following configuration in which 3d orbital participates in the FeH bond. 3
2
2
3d6 ð4s4pFe 1sH : a1 Þ ð3dFe 1sH : b2 Þ :
ð2Þ
Table 1 Equilibrium bond angle ðhe Þ of the lowest two triplet states and bond distances ðRe Þ of the triplet and quintet states Theory
he (°) 3 B1 3 A2
SBBb
MMPc
KUBd
KMLUTBe
101 101
– –
– –
– –
– –
1.688 1.672 – – –
1.653 1.635 1.634 – –
1.648 – – – –
1.665 – – – –
f
Re (A) 5 Dg 5 Pg 5 þ Rg 3 B1 3 A2 a
Experiment
Presenta
1.665 1.649 1.651 1.540 1.540
Results by MRSDCI + Q. Ref. [3]. c Ref. [6]. d Ref. [9]. e Ref. [7]. f Linear structure of the quintet states is established. b
K. Tanaka, K. Nobusada / Chemical Physics Letters 388 (2004) 389–394
Table 1 exhibits the calculated equilibrium bond distances, (Re ). The previous results are also included in the table. The ground state equilibrium distance of is in good agreement with the value observed 1.665 A by K€ orsgen and co-workers [7]. The equilibrium bond distances of the excited states, 5 Pg and 5 Rþ g , have not been observed. Judging from the level of approximation employed in the present work, the bond distances of 5 Pg and 5 Rþ g are expected to be close to those predicted values. The equilibrium bond distances of the triplet states are shorter than those of the quintet states, reflecting participation of 3d orbital in the FeH bond. 3.2. Term energies and lower vibration levels The term energy, Te (5 Pg 5 Dg ), obtained by the present calculation is 1861 cm1 and close to Marian’s value of 1783 cm1 (see Table 2). These values are close to the observed excitation energy of the anti-symmetric vibration, 1674 cm1 , of the ground state [9]. Since the excitation intensity of the anti-symmetric vibration should be stronger than that of the forbidden electronic excitation, 5 Pg 5 Dg , it is inferred that this situation makes it difficult to observe the T0 value for the latter transition.
Table 2 Term energies, Te (cm1 ), of the lowest two triplet states and the lowest two quintet states Theory
5
Pg 5 þ Rg 3 B1 3 A2 3
391
The term energies of the triplet states, Te (3 B1 X5 Dg ) and Te (3 A2 X5 Dg ), obtained by MRSDCI + Q were 1.16 104 cm1 (1.44 eV) and 1.22 104 cm1 (1.51 eV), respectively. It is noted that the 3 B1 , 3 A2 and 5 Dg states of FeH2 correlate to the lowest 4 D, 4 P and 6 D of FeH, among which the electron correlation in 4 D and 4 P is stronger than in 6 D [10]. The MRCPA method was applied to the 3 B1 , 3 A2 states and the ground state at their equilibrium geometries. As shown in Table 2, the predicted term energies, Te (3 B1 X5 Dg ) and Te (3 A2 X5 Dg ), were 1.00 104 cm1 (1.24 eV) and 1.06 104 cm1 (1.32 eV), respectively. They are about 0.2 eV lower than the MRSDCI + Q results. Because better agreement with experimental values is provided by MRCPA in FeH, we think these values are more reliable than those by MRSDCI + Q. Lower vibrational levels were calculated using the present adiabatic potential surfaces. Table 3 exhibits lower seven energy levels of the 5 Dg (X5 A1 , 5 B1 ), 5 5 Pg (5 A2 , 5 B2 ) and 5 Rþ g ((2) A1 ) states. The lowest four vibrational levels of the triplet states of FeH2 are shown in Table 4. The levels are designated by (mss ; mb ), where mss and mb are quantum number of the symmetric stretching and bending vibration modes, respectively. The (0,1)–(0,0) energy difference of the 5 Dg state of FeD2 , 272 or 269 cm1 , is close to Marian’s value of 250 cm1 [6] and they are comparable to the gas phase
Table 4 Lower bending vibrational levelsa and harmonic frequency (xss ) of the triplet states of FeH2 (in cm1 )
Presenta
SBBb
GPc
MMPd
(mss ; mb )b
3
1861 2695 1.00 104 1.06 104
2016 – – –
2134 2273 1.5 104 –
1783 2067 – –
(0,0) (0,1)–(0,0) (0,2)–(0,0) (0,3)–(0,0)
1063 328 693 1071
1162 464 927 1387
xss c
1798
1860
a 3 Results by MRSDCI + Q for 5 Pg , 5 Rþ g and by MRCPA(4) for B1 , A2 . b Ref. [3]. c Ref. [4]. d Ref. [6].
3
B1
A2
a
Same as (a) in Table 3. b Same as (b) in Table 3. c xss ¼ 3 ð0; 0Þ ð0; 1Þ.
Table 3 Lower vibration levelsa of the quintet states of FeH2 and FeD2 (in cm1 ) (mss ; mb )b
(0,0) (0,1)–(0,0) (0,2)–(0,0) (0,3)–(0,0) (0,4)–(0,0) (0,5)–(0,0) (1,0)–(0,0) a b
FeH2
FeD2
X5 A1
5
1078 379 765 1158 1556 1962 1821
1079 375 758 1147 1542 1943 1825
B1
5
A2
1061 343 700 1068 1445 1843 1834
B2
(2)5 A1
X5 A1
5
1205 620 1233 1839 2438 3031 1839
1029 291 613 956 1317 1697 1846
758 272 547 827 1109 1393 1247
759 269 542 819 1099 1381 1250
5
They are relative to the bottom of the potential surface of each state. mss : symmetric stretching, mb : bending.
B1
5
A2
739 245 498 756 1020 1289 1256
B2
(2)5 A1
845 450 896 1338 1776 2211 1253
714 203 424 659 904 1158 1259
5
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K. Tanaka, K. Nobusada / Chemical Physics Letters 388 (2004) 389–394
observed values 226 and 221 cm1 [8]. Those of FeH2 , 379 and 375 cm1 , are also close to Marian’s value of 354 cm1 [6]. Since unharmonicity is expected to be small, we calculated the bending energies, hxb , by using the (0,1)–(0,0) difference of the 3 B1 and 3 A2 states. They are 328 and 464 cm1 , respectively and almost the same order of those of the quintet states. The symmetric stretching vibration frequency, xss , was estimated assuming that the zero point energy of (0,0) is equal to hðxss þ xb Þ=2 and using hxb given by the (0,1)–(0,0) difference. The estimated values of the quintet states lie in the range of 1779 12 cm1 and those of 3 B1 and 3 A2 are 1798 and 1860 cm1 , respectively. 3.3. Non-adiabatic effects on the bending vibrational levels of the 5 B2 and (2)5 A1 states In the previous subsection, we neglected the nonadiabatic effect caused by a crossing between 5 B2 and (2)5 A1 (see Fig. 1). They both belong to the 5 A0 symmetry species when the molecule is distorted into the Cs point group. Owing to the crossing, the bending energy levels of these two states couple with each other through the anti-symmetric stretching vibration. The lowest six bending vibrational levels are presented in the figure, where the zero point energies are estimated by the half of the difference between (0,1) and (0,0) levels. Since these levels are close to the crossing point, it is desirable
Fig. 1. The bending potential curves of the 5 B2 (solid curve) and (2)5 A1 (broken curve) states and the lower bending vibrational levels: broken lines; the lowest three vibrational levels of 5 B2 and dotted lines; the lowest three vibrational level of (2)5 A1 .
to analyze the effect of non-adiabatic coupling. We derived a set of coupled equations of bending vibration, assuming the followings: (1) We assume that the symmetric and anti-symmetric stretching modes are not excited, because we are interested in the lower bending states and the frequencies of the both stretching modes are much larger than the bending frequencies. (2) Since the potential surfaces show that the variables of the symmetric stretching (FeH distance) and the bending (HFeH angle) are separable, the FeH distance of the symmetric motion, R, is kept constant (3.118 a.u. (1.650 A)). (3) We assume that the variable of the anti-symmetric mode is also separable from the other two modes. In the following discussion, we employ diabatic basis originating from 5 B2 and 5 A1 whose wavefunctions are expressed as wB2 and wA1 , respectively. Let us express wavefunctions of anti-symmetric stretching by /Bi 2 and 1 1 /A and bending wavefunctions by vBvb2 and vA j v0b . The 0 suffices i; j; vb ; and vb are vibrational quantum number. We consider the following two cases; wB2 /B0 2 vBvb2 perA1 A1 1 A1 turbed by wA1 /A 1 vv0b , and wA1 /0 vvb perturbed by B2 B2 wB2 /1 vv0 . When the anti-symmetric vibrational varib able is defined as r ¼ ðRFeH1 RFeH2 Þ=2 and the indices X and Y are designated as B2 or A1 , the coupling term for the bending motion is h/X1 jHXY j/Y0 i and further Y o/ j Y h/Xi jwX ow or j or i will contribute to the coupling. The Hamiltonian matrix element, HXY , is not zero when the molecule is distorted into the Cs symmetry. In this work, we consider only the former type of coupling which is discussed well [12] in NO2 . The equations and approximations are described in Appendix A. Table 5 shows the lower eight energy levels of FeH2 (the column represented by non-adiabatic). In addition to the column, the results without coupling are given in the column represented by diabatic. The two dimensional diabatic results (uncoupled between the two electronic states) are included to demonstrate how the variable separation is a reasonable approximation. The good agreement of one dimensional uncoupled results with the two dimensional ones indicates separability of the variables of bending and symmetric stretching. It is found that the non-adiabatic levels are shifted down by about 8–17 cm1 in comparison with the diabatic results. The shifts are independent of the relative separation of the levels. Table 6 shows overlap matrix elements between coupled wavefunctions and uncoupled (diabatic) wavefunctions. The wavefunctions are represented almost by a single wavefunction, although the mixing weights are gradually increasing in the higher levels. Very small energy shifts and weak mixing among the bending levels indicate that the nuclear bending motion of the two states is almost diabatic, as far as the lower levels concern.
K. Tanaka, K. Nobusada / Chemical Physics Letters 388 (2004) 389–394
393
Table 5 Non-adiabatic effect on the lower bending vibration through the conical intersection between 5 B2 and (2)5 A1 (in cm1 )a of FeH2 Coupled state number
Leading state
First Second Third Fourth Fifth Sixth Seventh Eighth
5
Two dimensionalb
Eqs. (A.1) and (A.2)
B2 B2 (2)5 A1 (2)5 A1 5 B2 (2)5 A1 (2)5 A1 5 B2 5
mb c
Non-adiabatic
Diabatic
Diabatic
0 1 0 1 2 2 3 3
303 922 954 1258 1536 1585 1931 2142
311 931 971 1272 1547 1600 1949 2159
310 930 974 1265 1543 1587 1931 2149
a
Relative to the bottom of the potential surfece of 5 B2 . From the results given by symmeric stretching and bending vibrations. c Bending quantum number. b
Table 6 Overlap between the coupled wavefunctions and diabatic wavefunctions for 5 B2 and (2)5 A1 of FeH2 Electronic state
5
mb a
B2
Coupled wavefunction First
Second
Third
Fourth
b
0 0.9984 0.0 )0.0005 0.0 )0.0001
0 0.0 0.9979 0.0 )0.0014 0.0
1 )0.1218 0.0 0.0157 0.0 0.0045
1 0.0 0.0922 0.0 )0.0246 0.0
ma b
1 0.0547 0.0 )0.0102 0.0 )0.0024
1 0.0 )0.0620 0.0 0.0160 0.0
0 0.9924 0.0 0.0019 0.0 0.0008
0 0.0 0.9954 0.0 )0.0008 0.0
ma 0 1 2 3 4
(2)5 A1 0 1 2 3 4 a b
Fifth
Sixth
Seventh
Eighth
0 0.0014 0.0 0.9967 0.0 0.0033
1 )0.0685 0.0 )0.0747 0.0 )0.0325
1 0.0 0.0495 0.0 )0.0360 0.0
0 0.0 0.0047 0.0 0.9933 0.0
1 )0.0319 0.0 ).0707 0.0 )0.0210
0 )0.0090 0.0 0.9941 0.0 )0.0039
0 0.0 )0.0043 0.0 0.9949 0.0
1 0.0 0.0766 0.0 0.0821 0.0
Bending quantum number. Antisymmetric stretching quantum number.
4. Conclusions The accurate adiabatic potential surfaces of the lowest two triplet states of FeH2 have been obtained by the MRSDCI + Q level of theory as well as the lowest five quintet states. The MRCPA method is also used to obtain accurate term energies of the triplet states. The lower symmetric stretching and bending vibration levels are calculated. The lowest two triplet states, 3 B1 and 3 A2 , are strongly bent in contrast to the lowest five quintet states. The electronic ground state is 5 Dg in agreement with the previous theoretical and experimental results. The triplet states lie above the ground state by 1.2–1.3 eV, whereas the quintet excited states are within 0.3 eV from the ground state. We have studied non-adiabatic coupling between the bending modes of the 5 B2 and (2)5 A1 states whose surfaces include a conical intersection. As far as the lower levels concern, the energy shifts caused by the coupling are very small and the nuclear motion of the two states
is almost diabatic. The shifts are independent of the relative separation between vibrational levels.
Acknowledgements This work was partially supported by a Grant-InAid for Scientific Research on a Priority Area (A), No. 12042201, from the Japanese Ministry of Education, Science, Sports, and Culture. We are grateful to a referee’s kind comments about non-adiabatic coupling.
Appendix A We derived the following set of coupled equations with the assumption shown in Section 3.3. The suffices 1 and 2 in the following two equations represent the 5 B2 or (2)5 A1 state. The suffix 1 indicates primary state and 2 means perturbing state.
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K. Tanaka, K. Nobusada / Chemical Physics Letters 388 (2004) 389–394
h2 d 2 þ W1 ðhÞ v1 ðhÞ þ V12 ðhÞv2 ðhÞ ¼ Ev1 ðhÞ; 2lR2 dh2 ðA:1Þ
2
2
the frequencies for the two electronic states. The value is nearly equal to that of the ground state. Fitting of V12 is carried out in the form V12 ¼ a þ bh2 þ ch4 :
h d þ hxa þ W2 ðhÞ v2 ðhÞ þ V21 ðhÞv1 ðhÞ ¼ Ev2 ðhÞ; 2lR2 dh2 ðA:2Þ
where h is defined as h ¼ ðp \HFeHÞ, W1 and W2 are adiabatic potential surfaces of the two states, and v1 and v2 are bending vibrational wavefunction of the two states. The reduced mass l is given by 2MH MFe ; ðA:3Þ 2MH þ MFe where MH and MFe are the mass of hydrogen and iron. The coupling term, V12 ð¼ V21 Þ, is defined as follows:
ðA:7Þ
a, b, and c are 0.000609, )0.000130, and 0.00103 in atomic unit for FeH2 . The surface crossing between the two states takes place at 10° of h by MRSDCI + Q calculation and near 27.4° by the MCSCF calculation. In applying the Eq. (A.7), we scaled the h by a factor 2.74, i.e. 2:74h. In solving Eqs. (A.1) and (A.2), we used a discrete variable representation [21].
l¼
ð1Þ ð2Þ V12 ¼ h/0 jH12 j/1 i; ð1Þ ð2Þ where /0 and /1 are
ðA:4Þ
the vibrational wavefunctions of the anti-symmetric stretching mode of the lowest level of the primary state and first excited level of the perturving state, respectively. H12 is a matrix element of the electronic Hamiltonian between the two electronic wavefunctions. Evaluation of H12 was carried out using the results of three states averaged MCSCF calculations for the 5 A0 states in Cs (X5 A1 , 5 B2 and (2)5 A1 in C2v ). The calculation was carried out at deformation variable, r, of 0.0, 0.025, 0.050, 0.075, 0.1, 0.15, 0.2 and 0.25 a.u. where R ¼ ðRFeH1 þ RFeH2 Þ=2 are kept to be 3.118 a.u. H12 was obtained from the following equation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA:5Þ DE ¼ DE02 þ 4H212 ; where DE and DE0 are energy separations between the second and third 5 A0 states at r and r ¼ 0, respectively. The matrix element H12 is fitted in the following form at each h (0°, 5°, 10°, 15°, 20°, 22°, 24°, 26°, 27°, 27.3°, 27.5°, 28°, 29°, 30°, 32°, 33°, and 40°) H12 ¼ ar þ br3 ;
ðA:6Þ
where a and b are determined by a least square technique. In evaluating V12 , we assumed harmonic vibrað1Þ ð2Þ tional wavefunctions for /0 and /1 with 1700 cm1 of
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