- Email: [email protected]
Distortion of interacting atoms and ions
Volume 12, number 2
CHEMICAL PHYSICS LETTERS
DISTORTION
OF INTERACTING,
15 Deozmber
1971
ATOMS. AND IONS
W.H. LAMS School of Chemistry, Rutgers University, New Brunswick, New Jersey, US.4 l and Chemistry Division, Argonne National Laboratory, Argonne, Illinois, USA ** mtd Lehrstuhl jib Theoretische Chemie der Technischen UniversitirtMlinchen, MUnchen, Germany
Received 16 June 1971 Revised manuscript received 17 September 1971
The distortion of atoms in HeNe and ions in LiF and NaCl is studied by use of non*rthogonal, Icast distorted. localized molecular orbit& in the restricted Hartree-Fock approximation. We find that the least distortion criterion is effective in producing localized orbitals. In addition we find that point-by-point the localized orbitalsdiffer little from the Hartree-Fock orbitais of the correvonding atoms or ions, although the differences are energetically important. We infer from our calculations that the effective potential which distorts atomic orbit& into lo~lized molecular orbitals must be quite weak.
This paper presents the first accurate,ly calculated restricted
Hartree-Fock
(KHF) localized molecular orbitals (L&IO’s) defmed by a least distortion criterion [l] . These calculations show that this criterion can be very effective and that the potential describing the interaction of an atom in a molecule with the
other parts of the molecule [2] must be weak. The latter point is quite important because we have found recently that potenfials of similar structure appear in a modified SciGdingBr equation for interacting atoms, ions, bonds, etc. [3], and in Lh40 equations in the multiconfiguration ielf-consistent field approximation [4]. It is these two ;pects which motivate our briefiy presenting what we regard as preliminary results. The non-orthogdnal LMO’s are defined in the followingway[1,2].jetQ=($1,$2,...,$N)bearow matrix of the occupied RHF molecular orbitals (MO) for a closed-shell molecule. Let @ = (&I ,&,..., :
#aiv,,~~,1,db2,...,~~~~,.-.)
be the mat+ of the I&IO’s for the closed-&Ii (ions, etc.) a, b, . .. &I the, tiolecule. .-
* Permanent address.
.
l*, Supported by’the U.S. Atomic Energy Commission.
xfla =N. The MO’s and LMO’s are reIated by the non-singular, square matrix C. Q=WZ.
(1)
For an isolated atom a the energy is a functionaI of the atomic orbitals, i.e.,_!?, = E, [Z,] , where 3 = (Qql >@a29.--,&Va)* The occupied RHF orbitals are thqse orthonormal orbjtals .$ = (&,&, ....&v& which m&e E, take the minimum value E,O=EQIL$], We require that the JLMO’sof atom a be orthonormal, satisfy (l), and make I?~[La] take the minimum value Ek. Thus in terms of the energy EG our NO’s Etre least distorted from the Q atom RHF orbiQfsQ**. We callEk - Ej the distortim energy. For closed-she11 atoms, ions, etc., the distortion energy is invariant under non-singular, linear transformations of the L&to’s_ Fdr this reason the distortion energy.may be regarded as an absolute measure of distortion. Note that the LMO’s for n are not orthogonal to the LMO’s of b, c, .: &t 8 [J&l & the rack operator for a as deter:
gtqqs *** T.L. GiI&rt has defined LMO’s which, under the above conditions, are ldst distorted from atom.i&orbita!s in an overkp sgse. .This is not equivalent to our aiterion (& ‘ref. [SJ). .’
Vplume-12.number 2
CHEMICAL Pi-iYSICS LETTERS
mined from E, [La]. The CtnP may be found by solving the Roothaan RHF equations
i2) in the standard way. The MO’s are the basis set for the LMO’s. Define _.
and v, = F - F,, where F is the Fock hamiltonian for the molecule. Then the LMO’s may be found without knowing the MO’s by solving the equations
for each atom, ion, etc., in the molecule [2]. It is pv,p which screens the potential v,. The effectiveness of this screening potential is tested by omcalcula: tions, as well as the effectiveness of the localization criterion, It should be clear from eq. (2) how one may analyse accurule RHF MO’s into LMO’i. One has only to solve the Roothaan equation for the atom, ion, etc., using the MO’s as a basis set. To this end we ‘have modified the BISON diatomic molecule program of Wahl and coworkers * 161. The one significant modification . we have made is to include subroutines to transform the integrals over Slater orbital sets into integrals over the MO’s. The procedure far doing this [7] is based on the procedure used in BISON fbi‘calculating efficiently the Coulomb and exchange matrices from the two-electron integrals. No integrals other than those used in the RHF MO calculations were needed. The accuracy of the LMO analysis is the same as that of the RHF MO calculations. When the integrals over the Slater basis set are available on tape, an LMO analysis for anyone of the systems we have studied was cornpleted’in about 5 minutes on an IBM 360/67. The molecuIes upon which LMCjanalyses have been carried out were chosen because each could be regarded as composed of two closed-shell subsystems, and because the integrals were available on tape at the Argonne National Laboratory. The RHF orbitals were * The BISON ref..[6]). 296
program is desuibed
fully by Wahl et al. (see
.,15December 1971
determined by Wahl and coworkers [8]. The HeNe and LiF orbital sets are each estimated to give a total energy within 0.01 au of the exact RHF energy [8]. For LiF the energy is only 0.004 au above the best value given in the literature [9]. The NaCl orbitals are less accurate, giving a total energy 0.025 au above the lowest value reported in the literature [9]. Even in this case the basis set was sufficiently flexible to allow the ions to distort. At Rutgers we have checked the calculations on these molecules, and we have also studied LiH and NaH. The results for LiH and NaH are not included here because H- is unstable in the RHF approximation. In table 1 the distorticn.energias are given. In calculating ,!$ only that part of the molecuIar basis set centered on CIwas used. The distortion energy for any atom or ion is,cluite small compared to its total energy, being at the largest less than 0.05% of the total energy. However the distortion energy for at least one atom or ion in each molecule is about 10% of the calculated binding energy for the molecule. The distortions are thus energetically small but not negligible. Only in the case of F- in LiF is the distortion energy larger than the estimated difference between the energies calculated with the MO’s used in this study and the RHF limit-. For this reason, these calculations will be repeated with MO’s giving energies within 0.001 au of the RHF limit when certain program modifications are completed. , .’ .: Although the distortion energits provide an absolute measure cf the degree to, which RHF atomic orbitals (AO) distort in forming molecules, they do not tell us how or where the orbitals distort. One way to study “how and where” is to draw graphs of the RHF AO’s and the corresponding LMO’s. We have done this for HeNe and NaCl:Here we present only. two of the graphs for HeNe since. they are.typical of what we have found, since the HeNe’MO’s are more delocalized than those of NaCl, and sin&Ne in HeNe appears to be more distorted than Cl-;in NaCl. Fig. I shows the two highest eneigy,, occu$ed o prbitals of HeNe as functions of the tiista& along the moiecular axis. Fig. 2 shows the 2~0 LMO of Ne in HeNe. In order to see clearly how the 2pa LMO differs from an Ne 2~0 AO, the scale of the graph must be increased along rhe vertical axis. The insert in fig. 2 pictures the behavior of the 2pa LMO near the.He nucleus but with the vertical scale blown up by a fac-
15 December 1971
CHEMICALPHYSICSLETTERS
Volume 12, number 2
Table 1
u
Binding
Basis size
MoIecub:
energy
Distortion
energy
b
0
n
R (au)
E-E,O-E;
Ea -.t$
Eb-
He
Ne
18
10
L? Na+
f;Cl-
t8 17
10 8
3.200 2.955 4.461
0.0213 -0.2920 [email protected]
0.0007 0.0010 0.0004
0.0024 0.0484 0.0124
z in au.
Fig. 1. The 30 and 40 MO’s of HeNe for points along the molecular (z) axis.The position of each nucleus is marked by a +.
many questions which they inspire. One question of this sort is how much of the point-by-point difference between the Ne 2po LMO and the Ne 2po A0 can be accounted for by mixing the Ne Is and 2s AO’s with the 2~0 AO. Another question is to what extent is the cusp in the Ne 2pa LMO at the He nucleus due to the He 1 so A0 _On the basis of a comparative study of RHF atomic valence-shell orbitals [I 1] we would not be surprised if nearly all of the cusp comes from the He lso AO. To answer such questions one needs a basis set independent method for the anaiysis of LMO’s into contributions from RHF AO’s of the various atoms in a molecule and from the part of Gilbert space not spanned
-2.0
0.0
2.0 z
4.0
6.0
in au.
Fk. 2. The Ne 2pu LMO in HeNe for points along the molecular (z) axis. This LMO is compared to the Ne 2~0 atomic orgital (AO) in the neighborhood of the He nucleus in an insert in which the vertical sule is blown up by a factor of 10. The position of each nucleus is marked by a +.
tor of 10. The distortion is small, but from the distortion energy we know that it is important. Thus the graphs supplement the information obtained from the distortion energies in the sense that we can see that the LMO’s are really not very different from AO’s. This should explain why it is advantageous to include atomic basis sets in a molecular basis set [lo]. Pi&ties of orbitals may inspire thought, but they do not give new numbers and they leave unanswered
E;
by the AO’s*.
Such a method
of
analysis has been developed and is being incorporated into our LMO program. Although this is a report of only the preliminary results of our planned study of the distortion of atoms ions, etc., in molecules, we have aL=ady learned some potentially important things. We have seen that our minimum distortion, localization criterion can be effective. How effectin! it is in any case can be regarded as an indication of the physical validity of a division of a molecule into particular kinds of interacting subsystems. From the fact that the LhlO’s differ so little from the corresponding RHF AO’s we infer that pvap in eq. (3) can be very effective in screening v=. This last point is very important because potentiaIs of the form v, - pv,,p appear in a modified SchrGdinger equation [3] for interacting atoms, etc., as well as in the L&l0 theory based on the multiconfiguration selfconsistent field approximation .[4] . This work was begun, and the initial programming * A basis set dependent method of analysis has been considered by Gilbert. r thank him for giving me a copy of his unpublished notes on this method in 1968.
Vqlume 12, number 2
15 December 1971
CkEMICAL PHYSICS LETTERS /
and computing done, while the author was a visiting scidntist at ,the Argonne National Laboratory in the summer of j968. I thank Dr. A.C. Wahl for inviting me there,, for letting me use his BISON program, and for sending me the IBM 360 version of BISON in .1969. Without BISON this work could not have been done. C&uIations at Rutgers have been supported by -the Rutgers Research CounCil and by grants of compuler time on Rutgers’ IBM 360/67. This paper was written in Miinchen, where I am supported in part by : the Technische UniversitCt and in part by Rutgers University. I thatik:Rofessor .G.L. Hofacker for inviting me here and for arranging for the support from the Technische.Universi&t, and the Rut&s Research Coimcil for making it possible for me to come.
298
‘References [I] W.H. Adams, .I. Cheni. Phys. 34 (1961) 89. [2] W.H. Adams, J:Chem. Phys. 37 (1962) 2009. [ 31 W.H. Adams, Chem. Phys. Letters 11 (1971) 441. [4j W.H. Adams, Chem. Phys. Letters 11 (1971) 71. [S] T.L. Gilbert, in: Molecular orbit& in chemistry, physics and biology, eds. P.-O. Lowdin and B. Pullman (Audemic Press, New York, 1964) p. 405. [6] AC. Wahl, P.J. Bertocini and R.H. Land, Argonne Natiqnal Laboratory Report No. 7271 (1968), unpublished. [7] A.C. W;lhl, private communication. (81 P. Suttnri, P. Bcrtocini, G. Das,T.L. Gilbert and A.C. .Wahl, Iatern. J. Quantum Chem. 3s (1970) 479. [9] A.D. McLean and M. Yoshimine, Tables of linear molecule wavefunctions (IBM Corporation, 1967). [lOI _ . P. Cade, K.D. Sales and A.C. Wahl, J. Chcm. Phys. 44 (1966) i973. [lit W.H. Adams, J. Am. Chem. Sot. 92 (1970) 2198.
CHEMICAL PHYSICS LETTERS
DISTORTION
OF INTERACTING,
15 Deozmber
1971
ATOMS. AND IONS
W.H. LAMS School of Chemistry, Rutgers University, New Brunswick, New Jersey, US.4 l and Chemistry Division, Argonne National Laboratory, Argonne, Illinois, USA ** mtd Lehrstuhl jib Theoretische Chemie der Technischen UniversitirtMlinchen, MUnchen, Germany
Received 16 June 1971 Revised manuscript received 17 September 1971
The distortion of atoms in HeNe and ions in LiF and NaCl is studied by use of non*rthogonal, Icast distorted. localized molecular orbit& in the restricted Hartree-Fock approximation. We find that the least distortion criterion is effective in producing localized orbitals. In addition we find that point-by-point the localized orbitalsdiffer little from the Hartree-Fock orbitais of the correvonding atoms or ions, although the differences are energetically important. We infer from our calculations that the effective potential which distorts atomic orbit& into lo~lized molecular orbitals must be quite weak.
This paper presents the first accurate,ly calculated restricted
Hartree-Fock
(KHF) localized molecular orbitals (L&IO’s) defmed by a least distortion criterion [l] . These calculations show that this criterion can be very effective and that the potential describing the interaction of an atom in a molecule with the
other parts of the molecule [2] must be weak. The latter point is quite important because we have found recently that potenfials of similar structure appear in a modified SciGdingBr equation for interacting atoms, ions, bonds, etc. [3], and in Lh40 equations in the multiconfiguration ielf-consistent field approximation [4]. It is these two ;pects which motivate our briefiy presenting what we regard as preliminary results. The non-orthogdnal LMO’s are defined in the followingway[1,2].jetQ=($1,$2,...,$N)bearow matrix of the occupied RHF molecular orbitals (MO) for a closed-shell molecule. Let @ = (&I ,&,..., :
#aiv,,~~,1,db2,...,~~~~,.-.)
be the mat+ of the I&IO’s for the closed-&Ii (ions, etc.) a, b, . .. &I the, tiolecule. .-
* Permanent address.
.
l*, Supported by’the U.S. Atomic Energy Commission.
xfla =N. The MO’s and LMO’s are reIated by the non-singular, square matrix C. Q=WZ.
(1)
For an isolated atom a the energy is a functionaI of the atomic orbitals, i.e.,_!?, = E, [Z,] , where 3 = (Qql >@a29.--,&Va)* The occupied RHF orbitals are thqse orthonormal orbjtals .$ = (&,&, ....&v& which m&e E, take the minimum value E,O=EQIL$], We require that the JLMO’sof atom a be orthonormal, satisfy (l), and make I?~[La] take the minimum value Ek. Thus in terms of the energy EG our NO’s Etre least distorted from the Q atom RHF orbiQfsQ**. We callEk - Ej the distortim energy. For closed-she11 atoms, ions, etc., the distortion energy is invariant under non-singular, linear transformations of the L&to’s_ Fdr this reason the distortion energy.may be regarded as an absolute measure of distortion. Note that the LMO’s for n are not orthogonal to the LMO’s of b, c, .: &t 8 [J&l & the rack operator for a as deter:
gtqqs *** T.L. GiI&rt has defined LMO’s which, under the above conditions, are ldst distorted from atom.i&orbita!s in an overkp sgse. .This is not equivalent to our aiterion (& ‘ref. [SJ). .’
Vplume-12.number 2
CHEMICAL Pi-iYSICS LETTERS
mined from E, [La]. The CtnP may be found by solving the Roothaan RHF equations
i2) in the standard way. The MO’s are the basis set for the LMO’s. Define _.
and v, = F - F,, where F is the Fock hamiltonian for the molecule. Then the LMO’s may be found without knowing the MO’s by solving the equations
for each atom, ion, etc., in the molecule [2]. It is pv,p which screens the potential v,. The effectiveness of this screening potential is tested by omcalcula: tions, as well as the effectiveness of the localization criterion, It should be clear from eq. (2) how one may analyse accurule RHF MO’s into LMO’i. One has only to solve the Roothaan equation for the atom, ion, etc., using the MO’s as a basis set. To this end we ‘have modified the BISON diatomic molecule program of Wahl and coworkers * 161. The one significant modification . we have made is to include subroutines to transform the integrals over Slater orbital sets into integrals over the MO’s. The procedure far doing this [7] is based on the procedure used in BISON fbi‘calculating efficiently the Coulomb and exchange matrices from the two-electron integrals. No integrals other than those used in the RHF MO calculations were needed. The accuracy of the LMO analysis is the same as that of the RHF MO calculations. When the integrals over the Slater basis set are available on tape, an LMO analysis for anyone of the systems we have studied was cornpleted’in about 5 minutes on an IBM 360/67. The molecuIes upon which LMCjanalyses have been carried out were chosen because each could be regarded as composed of two closed-shell subsystems, and because the integrals were available on tape at the Argonne National Laboratory. The RHF orbitals were * The BISON ref..[6]). 296
program is desuibed
fully by Wahl et al. (see
.,15December 1971
determined by Wahl and coworkers [8]. The HeNe and LiF orbital sets are each estimated to give a total energy within 0.01 au of the exact RHF energy [8]. For LiF the energy is only 0.004 au above the best value given in the literature [9]. The NaCl orbitals are less accurate, giving a total energy 0.025 au above the lowest value reported in the literature [9]. Even in this case the basis set was sufficiently flexible to allow the ions to distort. At Rutgers we have checked the calculations on these molecules, and we have also studied LiH and NaH. The results for LiH and NaH are not included here because H- is unstable in the RHF approximation. In table 1 the distorticn.energias are given. In calculating ,!$ only that part of the molecuIar basis set centered on CIwas used. The distortion energy for any atom or ion is,cluite small compared to its total energy, being at the largest less than 0.05% of the total energy. However the distortion energy for at least one atom or ion in each molecule is about 10% of the calculated binding energy for the molecule. The distortions are thus energetically small but not negligible. Only in the case of F- in LiF is the distortion energy larger than the estimated difference between the energies calculated with the MO’s used in this study and the RHF limit-. For this reason, these calculations will be repeated with MO’s giving energies within 0.001 au of the RHF limit when certain program modifications are completed. , .’ .: Although the distortion energits provide an absolute measure cf the degree to, which RHF atomic orbitals (AO) distort in forming molecules, they do not tell us how or where the orbitals distort. One way to study “how and where” is to draw graphs of the RHF AO’s and the corresponding LMO’s. We have done this for HeNe and NaCl:Here we present only. two of the graphs for HeNe since. they are.typical of what we have found, since the HeNe’MO’s are more delocalized than those of NaCl, and sin&Ne in HeNe appears to be more distorted than Cl-;in NaCl. Fig. I shows the two highest eneigy,, occu$ed o prbitals of HeNe as functions of the tiista& along the moiecular axis. Fig. 2 shows the 2~0 LMO of Ne in HeNe. In order to see clearly how the 2pa LMO differs from an Ne 2~0 AO, the scale of the graph must be increased along rhe vertical axis. The insert in fig. 2 pictures the behavior of the 2pa LMO near the.He nucleus but with the vertical scale blown up by a fac-
15 December 1971
CHEMICALPHYSICSLETTERS
Volume 12, number 2
Table 1
u
Binding
Basis size
MoIecub:
energy
Distortion
energy
b
0
n
R (au)
E-E,O-E;
Ea -.t$
Eb-
He
Ne
18
10
L? Na+
f;Cl-
t8 17
10 8
3.200 2.955 4.461
0.0213 -0.2920 [email protected]
0.0007 0.0010 0.0004
0.0024 0.0484 0.0124
z in au.
Fig. 1. The 30 and 40 MO’s of HeNe for points along the molecular (z) axis.The position of each nucleus is marked by a +.
many questions which they inspire. One question of this sort is how much of the point-by-point difference between the Ne 2po LMO and the Ne 2po A0 can be accounted for by mixing the Ne Is and 2s AO’s with the 2~0 AO. Another question is to what extent is the cusp in the Ne 2pa LMO at the He nucleus due to the He 1 so A0 _On the basis of a comparative study of RHF atomic valence-shell orbitals [I 1] we would not be surprised if nearly all of the cusp comes from the He lso AO. To answer such questions one needs a basis set independent method for the anaiysis of LMO’s into contributions from RHF AO’s of the various atoms in a molecule and from the part of Gilbert space not spanned
-2.0
0.0
2.0 z
4.0
6.0
in au.
Fk. 2. The Ne 2pu LMO in HeNe for points along the molecular (z) axis. This LMO is compared to the Ne 2~0 atomic orgital (AO) in the neighborhood of the He nucleus in an insert in which the vertical sule is blown up by a factor of 10. The position of each nucleus is marked by a +.
tor of 10. The distortion is small, but from the distortion energy we know that it is important. Thus the graphs supplement the information obtained from the distortion energies in the sense that we can see that the LMO’s are really not very different from AO’s. This should explain why it is advantageous to include atomic basis sets in a molecular basis set [lo]. Pi&ties of orbitals may inspire thought, but they do not give new numbers and they leave unanswered
E;
by the AO’s*.
Such a method
of
analysis has been developed and is being incorporated into our LMO program. Although this is a report of only the preliminary results of our planned study of the distortion of atoms ions, etc., in molecules, we have aL=ady learned some potentially important things. We have seen that our minimum distortion, localization criterion can be effective. How effectin! it is in any case can be regarded as an indication of the physical validity of a division of a molecule into particular kinds of interacting subsystems. From the fact that the LhlO’s differ so little from the corresponding RHF AO’s we infer that pvap in eq. (3) can be very effective in screening v=. This last point is very important because potentiaIs of the form v, - pv,,p appear in a modified SchrGdinger equation [3] for interacting atoms, etc., as well as in the L&l0 theory based on the multiconfiguration selfconsistent field approximation .[4] . This work was begun, and the initial programming * A basis set dependent method of analysis has been considered by Gilbert. r thank him for giving me a copy of his unpublished notes on this method in 1968.
Vqlume 12, number 2
15 December 1971
CkEMICAL PHYSICS LETTERS /
and computing done, while the author was a visiting scidntist at ,the Argonne National Laboratory in the summer of j968. I thank Dr. A.C. Wahl for inviting me there,, for letting me use his BISON program, and for sending me the IBM 360 version of BISON in .1969. Without BISON this work could not have been done. C&uIations at Rutgers have been supported by -the Rutgers Research CounCil and by grants of compuler time on Rutgers’ IBM 360/67. This paper was written in Miinchen, where I am supported in part by : the Technische UniversitCt and in part by Rutgers University. I thatik:Rofessor .G.L. Hofacker for inviting me here and for arranging for the support from the Technische.Universi&t, and the Rut&s Research Coimcil for making it possible for me to come.
298
‘References [I] W.H. Adams, .I. Cheni. Phys. 34 (1961) 89. [2] W.H. Adams, J:Chem. Phys. 37 (1962) 2009. [ 31 W.H. Adams, Chem. Phys. Letters 11 (1971) 441. [4j W.H. Adams, Chem. Phys. Letters 11 (1971) 71. [S] T.L. Gilbert, in: Molecular orbit& in chemistry, physics and biology, eds. P.-O. Lowdin and B. Pullman (Audemic Press, New York, 1964) p. 405. [6] AC. Wahl, P.J. Bertocini and R.H. Land, Argonne Natiqnal Laboratory Report No. 7271 (1968), unpublished. [7] A.C. W;lhl, private communication. (81 P. Suttnri, P. Bcrtocini, G. Das,T.L. Gilbert and A.C. .Wahl, Iatern. J. Quantum Chem. 3s (1970) 479. [9] A.D. McLean and M. Yoshimine, Tables of linear molecule wavefunctions (IBM Corporation, 1967). [lOI _ . P. Cade, K.D. Sales and A.C. Wahl, J. Chcm. Phys. 44 (1966) i973. [lit W.H. Adams, J. Am. Chem. Sot. 92 (1970) 2198.