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Many-body perturbation theory calculations on the electronic states of Li2, LiNa and Na2
CHEMICAL
Volume 81, number 2
MANY-BODY
PERTURBATION
ON THE ELECTRONIC
PHYSICS LETTERS
THEORY CALCULATIONS
STATES OF Liz, LiNa AND Na2
D-W. DAVIES and G.J.R. JONES* Department of Ckemrsrry. Universiry of Btkning~am, B&-mmg?ramB15 XT,
Uh*
Received 27 March 1981; in fmal form 10 April 1981
Quasi-degenerate many-body perturbation theory with a multi-configuration reference space is used to obtain potential curves for the ground and excited electronic states of Liz, LiNa and Naz. Correlation contributions are analyzed and the effect of potential curve crossing on laser action is discussed
l_ Introduction There are several ab initio calculations for the I xi ground states of I.$ and Na, and the l1: ground state of LiNa [I]. To obtain reliable potential curves for ground and excited states, it is clear that correlation must be included and MC!SCF potential curves for the A, B and X states of Na2 121, the X state of NaLi [3] and the X state of Liz [4] have been given by Wahl and co-workers_ Recently Konowalow et al. [S] calculated potential curves for the lowest 8 states of Na2 and Liz also by the MCSCF method. It is important to study different methods for potential curves, and many-body perturbation theory is a suitable technique _Kaldor [6] used this for 16 states of Hz, and Stern and Kaldor f7j have given results for 8 states of BH_ There has been a revival of theoretical interest in the alkali molecules due partly to their proposed use as lasers [5 j , and, from a general point of view, the position of Li, as the smallest stable molecule not involving hydrogen makes accurate determination of its electronic structure particularly important. We report here calculations on the lowest 1 xi* I%, 3Z,f,3Z;t, lI’I,, IfIg, 3IIu and 311g potential energy curves for I.$ and Naz , and the lowest two curves of each symmetry for the 1 Z+, 3 Z+, 1 II and * Present address: l)epartment of Physics, ‘University of w.rwick,Coventry
CV4 7AL, UK.
0 009--2614/81/0000-0000/$02.50
311 states of LiNa. The effect of the loss of inversion symmetry in LiNa makes the curves an interesting comparison with Ii2 and Na,.
2. Quasi-degeneratemany-body perturbation theory The approach used is the calculation of an effective hamiltonian matrix in a multidimensional model space of configurations by using the quasi-degenerate many-body perturbation theory (QD MBPT) due to Brandow [8 1, truncated to second_order. Kaldor [6] has used this approach, truncated at third order, with a one-dimensional model space, whilst Hegarty and Robb [9] have developed a procedure for partially summing diagrams and applied this with more general model spaces. QD MBPT is particularly suited to electronic structure calculations on small alkali metal systems owing to the clear separation between the energies of the core orbitals and the valency orbit&, making the core-valency distinction intrinsic to Brandow’s formulation unambiguous. Of the N electrons in the systems treated, N - 2 are considered to belong to the core, and all orbitals are calculated in the field of the core electrons ahowing the use of a PivB2 potential as the one-body part of the perturbation. This choice of perturbation and hence of the unperturbed ~~~to~~ results in the cancellation of certain diagrams, Let h^be the sum of the kinetic energy operator
0 North-Holland Publishing Company
279
CHEMICAL
Volume 8 1, number 2
PHYSICS LETTERS
and the electron-nucleus framework potential, then the unperturbed hamiltonian&u),in second quantized form, is given by
f?(O)=
C
(1)
C*?Cj=
I/
Ce,q+c, ,
(1)
I
where f$C, are creation and annihilation operators, respectively, and ei is the orbital energy of orbital i. The perturbation
(2) where the (ijjkl) are twoelectron repulsion integrals over spin orbitals. From an effective second-order hamiltonian, in Rayleigh-Schradinger theory, as shown by Brandow [S] , the core and valency diagrams given in fig. 1 are obtained [7] _In these calculations the model space consists of spin-addpted configurations involving only excitation of the two valency electrons, wrth maximum dimension 36 for given spatral and spin symmetry. The contracted gaussian orbital basis used for Li was that given by Davies and de1 Conde [lo] _ For Na, two p-orbitals, one with exponents 0.14461,0_04853 and contraction coefficients 0.23,0.56 respectively, and the other with exponent 0.01628 were added to
-,)I4 .I
._i
--
d.
Yr
Fig. I. The Brandow diagrams representing the effective interaction in QD hlBPT truncated at second order in the perturbation_ Diagrams (a) and (b) belong to the core set, (a) showing both first-order core diagrams, and (c)-(h) the valency set. Valency diagrams (d) and (e) refer to the effective one-body interaction and (c), (f)-(h) the effective two-body interaction. (tlxl), (ab),and (VI V, V3 t-4) refer to core, non-core and vaIency orbitals respectively.
280
15 July 1981
the contracted set No. 2 given by Veillard [ll]. All calculations were made on the UMRCC CDC 7600. The programme involved use of POLYATOM subroutines PA20, PA300 and PA4OC for the PN-2 calculations, transformation of the integrals with a MULTIBOND routine kindly supplied by Dr. G.G. Balint-Kurti, and calculation of the QD MBPT corrections with routines described elsewhere [ 12]_ A comparison was made with third-order non-degenerate results obtained from the programme written for IBM computers by Silver [13] and by Wilson [ 141, implemented on the CDC 7600, and with full valency CI results for the ground states of Li, and LiNa from the SPLICE programme on the IBM 370/165 at the University of Cambridge_ This provided a check on the QD MBPT results, and also material for a detailed discussion of the relative merits of the methods, which wrll be the subject of a further publication_
3 _Potential energy curves The calculated equilibrium bond lengths and dissociation energies for the states considered are given in table 1, and the T, values in table 2, with experirnental values for comparison. The predicted bond lengths for Li,, Na, and LiNa are longer than the experimental values, but the energies agree to within aO.2 eV, where experimental values are known. Calculations with full valency CI and third-order nondegenerate MBPT on the ground states of Li, and LiNa with the same basis give very similar bond lengths to those reported here _The long predicted bond lengths probably result from the neglect of angular correlation about the internuclear axis, an inherent limitation of the basis sets used, this form of correlation being of particular importance at smaller internuclear separations_ Confirmation of this was provided by the use of a single p-type contracted gaussian orbital on Li, for which full valency CI and thirdorder MBPT produce a bond length of over 5.4 au. Theoretical predictions using MC SCF [2,5,26] and pseudopotential approaches have generally overestimated bond lengths and underestimated dissociation energies, even with larger basis sets than those used here _Semiempirical pseudopotential calculations have in addition incorporated core-valency conelation [27-291 unlike the MC SCF calculations_ CEPA
Volume 81, number 2
CHEMICAL
PHYSICS
15 July 1981
LETTERS
Table 1 Calculated values for dissociation energies De (in eV) and bond lengths Re (in bohr). Experimental values are in brackets. State
State
Naz
LIZ De
Re
De
X’Z* g
0.803 (1.05)“)
5 -16 (5 05) b)
0.650 (0.742) e,
A’.Z+ U
1.034 (1.11)b)
5 90 (5 88) c,
0 947 (0.949,1.026)
0.027 (0.38) d,
5.81 (5 56)d)
0.002 (0.328) f)
Bll-Iu
Re
e,f)
L&a De
Re
0.735 (0.928, o_9o)g*h)
556 (5 -34) h)
5.95 (5.82) e,
xlZZ+
7.00 (6.88)f)
A’-?+ -
6.81 (6.45) f,
B’I-I
0.120 i) (0.12) h)
0.980
6.45 9.47 1) (6.10) h)
C’Fig
0.136
8.88
0.124
10.10
C1n
o_ooo
6.96
a3rf
0.014
10.00
0 032
1059
a3r+
0.022
10 25
b3gu
1.293
5.03
1 007
6 08
b3n
C3Z+
0.713
5.92
0.441
7.70
.=3X+
1.146 0.574
5.55 6 60
82
y)Ref_[15]. b)Ref.[16]. C)Ref_[17]. d)Ref.[18]. e)Ref.[19]. ‘) Long-range minimum: no calculated short-range minimum exists. calculations [30] on the ground state of LiNa including core-valency correlation give more accurate dipole moment, bond length and other molecular constants than those with valency-valency correlation only. In the present work core-vaIency effects are incorporated at second order in the perturbation, and
f)Ref.[20].
g)Ref.[21].
Il)Ref
[22]
discussed [5,29,3 l-341 - Both A-X and B-X band laser action for Li2 and Na2 have been observed [32-
34]_ York and Gallagher [31] have discussed the conditions for a high-powered A-X band laser in a
the method in general seems promising. The form of the potential energy curves is clearly important when the use of alkali dimers as lasers is Table 2 Calculated T, values (in eV) for six excited states. Experimental values are in brackets State
X’Z+ A’$ Bin, C1ng
Lip
Naz
Te
Te
0.0 1.686 (1.744~) 2691 (22534) b, 2.595
a3Z+ U
0.789
b31-Iu
1.429
C3Z+
2.006
E
A) Ref. [23]. e, Ref. [25].
State
LiNNa Te
0.0 1.627 (1820) c) 2572 (2519)d) 2.450
X’Z+ A’P
0.0 1.671
B1n
2596 (25i)f)
cln
2.659
0.619
a3Z+
0.713
1.568 (< 1.820)e)
b3n
1505
CJZ+
2.075
b) Ref_ [lS]. f, Ref. 1221.
2.134
Cl Ref. [20].
d, Ref. 1241.
-14_86-
-14.904-o
8.0
12.0
RISohr) Fig. 2. Liz - potential curves of eight states. 281
1.5 July 1981
CHEMICAL PHYSICS LETTERS
Volume 81, number 2
out that crossing of the curves and 3 Zz states provides a mechanism for predissociation with depopulation of the A state. In agreement with the MC SCF calculations of Konowalow and Olson [26], fig. 2 predicts such a crossing for Li, , but fig. 3 shows for Na2 the possibility of crossing only high on the respulsive limb of the plasma, and pointed
for the 3 ll,
-323.83
curve for a 311u with no impairment
c
of laser action.
-323.87
z G ki 5
4. Core and valency correlation
-323.91
-3 23.95
sfo
9Io
13Io
R Eohr) Fig. 3. Naa - potential curves of eight states_
Table 3 shows the differences between energies obtamed from effective hamiltonians truncated at first and second order in the perturbation, defied in the same model space, divided into contributions from core and valency diagrams. The secondorder core diagram in association with the perturbation given by eq. (2) corresponds to the secondorder correlatron energy of the ground state of the physical core (i.e. Liz+ L,iNa2+ Na$+) rather than the core of the neutral molecules,
and contains
exclusion
principle
violating
(EPV) terms which cancel EPV diagrams in the valency set_ From table 3 the total second-order correlation energy in QD MBPT of the ground state of Liz at 5 20 au is 0.0119 au, and that of LiNa at 5 50 au is 0 .1330 au _These closely match the values for the sum of core-core, and core-valency correlation energies of 0.0118 au, and 0.1333 au respectively, obtained by Table 3 Seeondarder correction to correlation energy of eight states of Lis and Nas _Contribution of core and valency diagrams (in au X 104) 2=5 core diagram
-169.39
-16 9-43 8:0 R (Bohr) Fig. 4. LiNa - potential curves of eight states.
282
112.15
valency diagrams X’Z+ 7.07 Alrg 4.79 BQ-I,U 6.96 C’rlg 456 aa*u 5.41 b3rI, 5 -40 C3Pg 5.17 d3ng 5.27
20)
$6.00)
LiNa (R =5.50)
253251
132251
9.27 4.43 6.19 3.36 8.10 3.36 2.74 452
X1x+ A ‘Z+ B’I-I C’H a3C+ b3n csz+ d3n
8.24 4.75 623 432 6.79 456 399 4.89
Volume
81, number
CHEMICAL
2
PHYSICS
adding the appropriate pair energies associated with a second-order nondegenerate perturbation calculation, using a ground-state Hartree-Fock unperturbed wavefunction and Mgller-Plesset partitioning of the hamiltonian [12] _It is apparent-that valency-valency correlation is fully accounted for in the first-order effective hamiltonian leaving only core-core and core-valency effects in the second (and higher) orders of QD MBPT. That is to be expected, as the first-order effective hamiltonian matrix is essentially a full valency configuration interaction matrix, differing only m the absence of configurations involving high energy orbitals. Since the basic form of the potential energy curves, at higher accuracy than the independent-particle model, is determined by valency-valency correlation, the mfluence of the second-order terms in the effective hamiltonian is small in this work. Core-valency effects, which enter here at second order, may, however, be rmportant for the dipole moment of LiNa and other properties of the alkali diatomic molecules.
Acknowledgement S.R.C. is thanked for an award to GJ.RJ.
[l] W.G. Richards, TE.H_ Walker and RX.
Hinkley. A brbbowavefunctions (Clarendon
graphy of ab initio molecular Press, Oxord, 1971); W-G. Richards, T.E.H. Walker, L. Farnell and P.R. Scott, A brbriography of ab initio mote&es wavefunctions, Supplement 1970-1973 (Ciarendon Press, Oxford, 1973); W.G. Richards, P.R. Scott, E.A. Colbourne and A-F. Marchington, A bibliography of ab initio molecules wavefunctions, Supplement 1974-1977 (Clarendon Press,
[3] [4]
Oxford, 1978). W J. Stevens, MM_ Hessel, P J. Bertoneini and A C. Wahl, J. Chem. Phys. 66 (1977) 1477. PJ. Bertoncini, G. Das and AC. Wahl, J. Chem. Phys. 52 (1970) 5 112. P. Sutton, PJ. Bertoncini, G. Das, T.L. Gilbert and AC. Wahl, Intern. J. Quantum Chem. 3s (1970) 479.
15 July
1981
[S] D.D. Konowalow. M.E. Rosenkrantz and M.L. Olson, J. Chem. Phys. 72 (1980) 2612. 161U. Kaldor, J. Chem. Phys. 63 (1975) 2199. r71 P.S. Stern and U. Kaldor, J. Chem. Phys. 64 (1976) 2002. 181 B.H. Brandow, Rev. Mod. Phys. 39 (1967) 771, Advan. Quantum Chem. 10 (1977) 187. [91 D. Hegarty and MA_ Robb, Mel_ Phys. 37 (1979) 1455. 1101 D-W. Davies and G. de1 Conde, Chem. Phys 12 (1976) 45. 1111 A. Verllard, Theoret_ Chim. Acta 12 (1968) 40.5. [I21 G J-R. Jones, Ph.D. Tbesrs, University of Birmingham (1980). 1131 DM_ Srlver, Computer Phys. Commun. 14 (1978) 71,Sl. 1141 S. Wilson, Computer Phys Commun. 14 (1978) 91. [ISI R. Velasco. C.H. Ottrnger and R.N. Zare, J. Chem. Phys. 51 (1969) 5522. I161 P Kusch and MM. Hessel, J. Chem. Phys. 67 (1977)586_ r171 R. Velasco and F. Rivero, Optrca Pura Applicada S (1972) 76. [ISI M.M. Hessel and C.R. Vidal, J. Chem. Phys. 70 (1979) 4439. 1191 W. Demtrgder and M. Stock, J. Mol. Spectry. 55 (1975) 476. P. Kusch and MM. Hessel, J. Chem. Phys. 68 (1978) 2591. PI P11 K.F. Zmbov, C H. Wu and H.R. Ihle, J. Chem. Phys 67 (1977) 4603. WI MM_ Hessel, Phys. Rev. Letters 26 (1971) 215. [23] G M. Almy and G R. Irwin, Phys. Rev. 49 (1936) 72. [24] W. Demtrbder, M. McLintock and RN_ Zare, J. Chem. Phys 51 (1969) 5495. [2-S] K-P. Huber and G. Herzberg, Constants of diatomic
WI
References
[2]
LETTERS
1271 PI WI 1301 1311
1321
[331 1341
molecules (van Nostrand, Princeton, 1979). D-D. Konowalow and M.L. Olson, I_ Chem. Phys 71 (1979) 450. D-K. Watson, CJ. Cejan, S. Guberman and A. Dalgarno, Chem. Phys. Letters50 (1977) 181. P. Habitz, W.H.E. Schwartz and R. Ahirrchs, J. Chem. Phys. 66 (1977) 5117. J-N. Bardsley, B.R. Junker and D.W. Norcross,Chem. Phys. Letters 37 (1976) 502. R. Rosmus and W! Meyer, J. Chem. Phys. 65 (1976) 492. G. York and A. Gallagher, Joint Institute for Laboratory Astrophysics, Boulder, Colorado, JILA Report 114 (1974). H. Wellmg and B. Wellgehausen, in: Proceedings of the Third Conference on Laser Spectroscopy, eds. J.L. Hall and J.L. Carlsten (Springer, Berlin, 1977) p. 365. B. Wellgehausen, K.H_ Stephan, D. Friede and H. Welling, Opt. Commun. 23 (1977) 157. MA. Henesian, R.L. Herbst and R.L. Byer, J. Appl. Phys. 47 (1976) 1515.
283
Volume 81, number 2
MANY-BODY
PERTURBATION
ON THE ELECTRONIC
PHYSICS LETTERS
THEORY CALCULATIONS
STATES OF Liz, LiNa AND Na2
D-W. DAVIES and G.J.R. JONES* Department of Ckemrsrry. Universiry of Btkning~am, B&-mmg?ramB15 XT,
Uh*
Received 27 March 1981; in fmal form 10 April 1981
Quasi-degenerate many-body perturbation theory with a multi-configuration reference space is used to obtain potential curves for the ground and excited electronic states of Liz, LiNa and Naz. Correlation contributions are analyzed and the effect of potential curve crossing on laser action is discussed
l_ Introduction There are several ab initio calculations for the I xi ground states of I.$ and Na, and the l1: ground state of LiNa [I]. To obtain reliable potential curves for ground and excited states, it is clear that correlation must be included and MC!SCF potential curves for the A, B and X states of Na2 121, the X state of NaLi [3] and the X state of Liz [4] have been given by Wahl and co-workers_ Recently Konowalow et al. [S] calculated potential curves for the lowest 8 states of Na2 and Liz also by the MCSCF method. It is important to study different methods for potential curves, and many-body perturbation theory is a suitable technique _Kaldor [6] used this for 16 states of Hz, and Stern and Kaldor f7j have given results for 8 states of BH_ There has been a revival of theoretical interest in the alkali molecules due partly to their proposed use as lasers [5 j , and, from a general point of view, the position of Li, as the smallest stable molecule not involving hydrogen makes accurate determination of its electronic structure particularly important. We report here calculations on the lowest 1 xi* I%, 3Z,f,3Z;t, lI’I,, IfIg, 3IIu and 311g potential energy curves for I.$ and Naz , and the lowest two curves of each symmetry for the 1 Z+, 3 Z+, 1 II and * Present address: l)epartment of Physics, ‘University of w.rwick,Coventry
CV4 7AL, UK.
0 009--2614/81/0000-0000/$02.50
311 states of LiNa. The effect of the loss of inversion symmetry in LiNa makes the curves an interesting comparison with Ii2 and Na,.
2. Quasi-degeneratemany-body perturbation theory The approach used is the calculation of an effective hamiltonian matrix in a multidimensional model space of configurations by using the quasi-degenerate many-body perturbation theory (QD MBPT) due to Brandow [8 1, truncated to second_order. Kaldor [6] has used this approach, truncated at third order, with a one-dimensional model space, whilst Hegarty and Robb [9] have developed a procedure for partially summing diagrams and applied this with more general model spaces. QD MBPT is particularly suited to electronic structure calculations on small alkali metal systems owing to the clear separation between the energies of the core orbitals and the valency orbit&, making the core-valency distinction intrinsic to Brandow’s formulation unambiguous. Of the N electrons in the systems treated, N - 2 are considered to belong to the core, and all orbitals are calculated in the field of the core electrons ahowing the use of a PivB2 potential as the one-body part of the perturbation. This choice of perturbation and hence of the unperturbed ~~~to~~ results in the cancellation of certain diagrams, Let h^be the sum of the kinetic energy operator
0 North-Holland Publishing Company
279
CHEMICAL
Volume 8 1, number 2
PHYSICS LETTERS
and the electron-nucleus framework potential, then the unperturbed hamiltonian&u),in second quantized form, is given by
f?(O)=
C
(1)
C*?Cj=
I/
Ce,q+c, ,
(1)
I
where f$C, are creation and annihilation operators, respectively, and ei is the orbital energy of orbital i. The perturbation
(2) where the (ijjkl) are twoelectron repulsion integrals over spin orbitals. From an effective second-order hamiltonian, in Rayleigh-Schradinger theory, as shown by Brandow [S] , the core and valency diagrams given in fig. 1 are obtained [7] _In these calculations the model space consists of spin-addpted configurations involving only excitation of the two valency electrons, wrth maximum dimension 36 for given spatral and spin symmetry. The contracted gaussian orbital basis used for Li was that given by Davies and de1 Conde [lo] _ For Na, two p-orbitals, one with exponents 0.14461,0_04853 and contraction coefficients 0.23,0.56 respectively, and the other with exponent 0.01628 were added to
-,)I4 .I
._i
--
d.
Yr
Fig. I. The Brandow diagrams representing the effective interaction in QD hlBPT truncated at second order in the perturbation_ Diagrams (a) and (b) belong to the core set, (a) showing both first-order core diagrams, and (c)-(h) the valency set. Valency diagrams (d) and (e) refer to the effective one-body interaction and (c), (f)-(h) the effective two-body interaction. (tlxl), (ab),and (VI V, V3 t-4) refer to core, non-core and vaIency orbitals respectively.
280
15 July 1981
the contracted set No. 2 given by Veillard [ll]. All calculations were made on the UMRCC CDC 7600. The programme involved use of POLYATOM subroutines PA20, PA300 and PA4OC for the PN-2 calculations, transformation of the integrals with a MULTIBOND routine kindly supplied by Dr. G.G. Balint-Kurti, and calculation of the QD MBPT corrections with routines described elsewhere [ 12]_ A comparison was made with third-order non-degenerate results obtained from the programme written for IBM computers by Silver [13] and by Wilson [ 141, implemented on the CDC 7600, and with full valency CI results for the ground states of Li, and LiNa from the SPLICE programme on the IBM 370/165 at the University of Cambridge_ This provided a check on the QD MBPT results, and also material for a detailed discussion of the relative merits of the methods, which wrll be the subject of a further publication_
3 _Potential energy curves The calculated equilibrium bond lengths and dissociation energies for the states considered are given in table 1, and the T, values in table 2, with experirnental values for comparison. The predicted bond lengths for Li,, Na, and LiNa are longer than the experimental values, but the energies agree to within aO.2 eV, where experimental values are known. Calculations with full valency CI and third-order nondegenerate MBPT on the ground states of Li, and LiNa with the same basis give very similar bond lengths to those reported here _The long predicted bond lengths probably result from the neglect of angular correlation about the internuclear axis, an inherent limitation of the basis sets used, this form of correlation being of particular importance at smaller internuclear separations_ Confirmation of this was provided by the use of a single p-type contracted gaussian orbital on Li, for which full valency CI and thirdorder MBPT produce a bond length of over 5.4 au. Theoretical predictions using MC SCF [2,5,26] and pseudopotential approaches have generally overestimated bond lengths and underestimated dissociation energies, even with larger basis sets than those used here _Semiempirical pseudopotential calculations have in addition incorporated core-valency conelation [27-291 unlike the MC SCF calculations_ CEPA
Volume 81, number 2
CHEMICAL
PHYSICS
15 July 1981
LETTERS
Table 1 Calculated values for dissociation energies De (in eV) and bond lengths Re (in bohr). Experimental values are in brackets. State
State
Naz
LIZ De
Re
De
X’Z* g
0.803 (1.05)“)
5 -16 (5 05) b)
0.650 (0.742) e,
A’.Z+ U
1.034 (1.11)b)
5 90 (5 88) c,
0 947 (0.949,1.026)
0.027 (0.38) d,
5.81 (5 56)d)
0.002 (0.328) f)
Bll-Iu
Re
e,f)
L&a De
Re
0.735 (0.928, o_9o)g*h)
556 (5 -34) h)
5.95 (5.82) e,
xlZZ+
7.00 (6.88)f)
A’-?+ -
6.81 (6.45) f,
B’I-I
0.120 i) (0.12) h)
0.980
6.45 9.47 1) (6.10) h)
C’Fig
0.136
8.88
0.124
10.10
C1n
o_ooo
6.96
a3rf
0.014
10.00
0 032
1059
a3r+
0.022
10 25
b3gu
1.293
5.03
1 007
6 08
b3n
C3Z+
0.713
5.92
0.441
7.70
.=3X+
1.146 0.574
5.55 6 60
82
y)Ref_[15]. b)Ref.[16]. C)Ref_[17]. d)Ref.[18]. e)Ref.[19]. ‘) Long-range minimum: no calculated short-range minimum exists. calculations [30] on the ground state of LiNa including core-valency correlation give more accurate dipole moment, bond length and other molecular constants than those with valency-valency correlation only. In the present work core-vaIency effects are incorporated at second order in the perturbation, and
f)Ref.[20].
g)Ref.[21].
Il)Ref
[22]
discussed [5,29,3 l-341 - Both A-X and B-X band laser action for Li2 and Na2 have been observed [32-
34]_ York and Gallagher [31] have discussed the conditions for a high-powered A-X band laser in a
the method in general seems promising. The form of the potential energy curves is clearly important when the use of alkali dimers as lasers is Table 2 Calculated T, values (in eV) for six excited states. Experimental values are in brackets State
X’Z+ A’$ Bin, C1ng
Lip
Naz
Te
Te
0.0 1.686 (1.744~) 2691 (22534) b, 2.595
a3Z+ U
0.789
b31-Iu
1.429
C3Z+
2.006
E
A) Ref. [23]. e, Ref. [25].
State
LiNNa Te
0.0 1.627 (1820) c) 2572 (2519)d) 2.450
X’Z+ A’P
0.0 1.671
B1n
2596 (25i)f)
cln
2.659
0.619
a3Z+
0.713
1.568 (< 1.820)e)
b3n
1505
CJZ+
2.075
b) Ref_ [lS]. f, Ref. 1221.
2.134
Cl Ref. [20].
d, Ref. 1241.
-14_86-
-14.904-o
8.0
12.0
RISohr) Fig. 2. Liz - potential curves of eight states. 281
1.5 July 1981
CHEMICAL PHYSICS LETTERS
Volume 81, number 2
out that crossing of the curves and 3 Zz states provides a mechanism for predissociation with depopulation of the A state. In agreement with the MC SCF calculations of Konowalow and Olson [26], fig. 2 predicts such a crossing for Li, , but fig. 3 shows for Na2 the possibility of crossing only high on the respulsive limb of the plasma, and pointed
for the 3 ll,
-323.83
curve for a 311u with no impairment
c
of laser action.
-323.87
z G ki 5
4. Core and valency correlation
-323.91
-3 23.95
sfo
9Io
13Io
R Eohr) Fig. 3. Naa - potential curves of eight states_
Table 3 shows the differences between energies obtamed from effective hamiltonians truncated at first and second order in the perturbation, defied in the same model space, divided into contributions from core and valency diagrams. The secondorder core diagram in association with the perturbation given by eq. (2) corresponds to the secondorder correlatron energy of the ground state of the physical core (i.e. Liz+ L,iNa2+ Na$+) rather than the core of the neutral molecules,
and contains
exclusion
principle
violating
(EPV) terms which cancel EPV diagrams in the valency set_ From table 3 the total second-order correlation energy in QD MBPT of the ground state of Liz at 5 20 au is 0.0119 au, and that of LiNa at 5 50 au is 0 .1330 au _These closely match the values for the sum of core-core, and core-valency correlation energies of 0.0118 au, and 0.1333 au respectively, obtained by Table 3 Seeondarder correction to correlation energy of eight states of Lis and Nas _Contribution of core and valency diagrams (in au X 104) 2=5 core diagram
-169.39
-16 9-43 8:0 R (Bohr) Fig. 4. LiNa - potential curves of eight states.
282
112.15
valency diagrams X’Z+ 7.07 Alrg 4.79 BQ-I,U 6.96 C’rlg 456 aa*u 5.41 b3rI, 5 -40 C3Pg 5.17 d3ng 5.27
20)
$6.00)
LiNa (R =5.50)
253251
132251
9.27 4.43 6.19 3.36 8.10 3.36 2.74 452
X1x+ A ‘Z+ B’I-I C’H a3C+ b3n csz+ d3n
8.24 4.75 623 432 6.79 456 399 4.89
Volume
81, number
CHEMICAL
2
PHYSICS
adding the appropriate pair energies associated with a second-order nondegenerate perturbation calculation, using a ground-state Hartree-Fock unperturbed wavefunction and Mgller-Plesset partitioning of the hamiltonian [12] _It is apparent-that valency-valency correlation is fully accounted for in the first-order effective hamiltonian leaving only core-core and core-valency effects in the second (and higher) orders of QD MBPT. That is to be expected, as the first-order effective hamiltonian matrix is essentially a full valency configuration interaction matrix, differing only m the absence of configurations involving high energy orbitals. Since the basic form of the potential energy curves, at higher accuracy than the independent-particle model, is determined by valency-valency correlation, the mfluence of the second-order terms in the effective hamiltonian is small in this work. Core-valency effects, which enter here at second order, may, however, be rmportant for the dipole moment of LiNa and other properties of the alkali diatomic molecules.
Acknowledgement S.R.C. is thanked for an award to GJ.RJ.
[l] W.G. Richards, TE.H_ Walker and RX.
Hinkley. A brbbowavefunctions (Clarendon
graphy of ab initio molecular Press, Oxord, 1971); W-G. Richards, T.E.H. Walker, L. Farnell and P.R. Scott, A brbriography of ab initio mote&es wavefunctions, Supplement 1970-1973 (Ciarendon Press, Oxford, 1973); W.G. Richards, P.R. Scott, E.A. Colbourne and A-F. Marchington, A bibliography of ab initio molecules wavefunctions, Supplement 1974-1977 (Clarendon Press,
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Oxford, 1978). W J. Stevens, MM_ Hessel, P J. Bertoneini and A C. Wahl, J. Chem. Phys. 66 (1977) 1477. PJ. Bertoncini, G. Das and AC. Wahl, J. Chem. Phys. 52 (1970) 5 112. P. Sutton, PJ. Bertoncini, G. Das, T.L. Gilbert and AC. Wahl, Intern. J. Quantum Chem. 3s (1970) 479.
15 July
1981
[S] D.D. Konowalow. M.E. Rosenkrantz and M.L. Olson, J. Chem. Phys. 72 (1980) 2612. 161U. Kaldor, J. Chem. Phys. 63 (1975) 2199. r71 P.S. Stern and U. Kaldor, J. Chem. Phys. 64 (1976) 2002. 181 B.H. Brandow, Rev. Mod. Phys. 39 (1967) 771, Advan. Quantum Chem. 10 (1977) 187. [91 D. Hegarty and MA_ Robb, Mel_ Phys. 37 (1979) 1455. 1101 D-W. Davies and G. de1 Conde, Chem. Phys 12 (1976) 45. 1111 A. Verllard, Theoret_ Chim. Acta 12 (1968) 40.5. [I21 G J-R. Jones, Ph.D. Tbesrs, University of Birmingham (1980). 1131 DM_ Srlver, Computer Phys. Commun. 14 (1978) 71,Sl. 1141 S. Wilson, Computer Phys Commun. 14 (1978) 91. [ISI R. Velasco. C.H. Ottrnger and R.N. Zare, J. Chem. Phys. 51 (1969) 5522. I161 P Kusch and MM. Hessel, J. Chem. Phys. 67 (1977)586_ r171 R. Velasco and F. Rivero, Optrca Pura Applicada S (1972) 76. [ISI M.M. Hessel and C.R. Vidal, J. Chem. Phys. 70 (1979) 4439. 1191 W. Demtrgder and M. Stock, J. Mol. Spectry. 55 (1975) 476. P. Kusch and MM. Hessel, J. Chem. Phys. 68 (1978) 2591. PI P11 K.F. Zmbov, C H. Wu and H.R. Ihle, J. Chem. Phys 67 (1977) 4603. WI MM_ Hessel, Phys. Rev. Letters 26 (1971) 215. [23] G M. Almy and G R. Irwin, Phys. Rev. 49 (1936) 72. [24] W. Demtrbder, M. McLintock and RN_ Zare, J. Chem. Phys 51 (1969) 5495. [2-S] K-P. Huber and G. Herzberg, Constants of diatomic
WI
References
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LETTERS
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