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A valence-bond analysis of the improved bond-orbital wavefunction
9 Apnl 1992
CHEhlICAL PHYSICS LEllXRS
Volume 87. number 5
A VALENCE-BOND
ANALYSIS
Valerio
and Gian Franc0 MUSS0
MAGNASCO
OF THE IMPROVED
BOND-ORBITAL
WAVEFUNCTION
Isritulo di Chimim Indusniale dell’llnrrersit~. 16132 Genw, Ita!,
Received 25 Jsnuary 1982
An anzdysis ol the unproved bond-orbnal wwclunction m terms of localizrd singletVB structures reveals the bondmg and chug transfer occurring between the four electrons wluch arc uwolrcd in each excitation. The phywal nature oi the undcrlymg approGrn3tion can hence be dlscussed.
1. Introduction We recently
[I,?]
proposed an improved
to localized doubly
antlbondiig
occupied
(o*) orbitals
to second order. Prehminary [3,4].
theory.
approach by Including
polanzation
and delocahza-
Singlet coupled configurations
dercribmg
sit@ excnmons
are allowed to mteract with the one-configuratlon
localized bondmg (u) orbitals applications
ergy results and wavefunctions
(90)
bond-orbital
tion effects in the second order of perturbation
through a non-orthogonal
to intermolecular
[l]
[6-81
to an extent determined
seemingly approximated
excitation.
SCF WF does include
[7_] problems give cn-
Induction
by the stze of its basis set, and in our perturbative
(VB) structures
More precisely,
the nature of the resultmg many-electron
concerning
the pairs of two-electron
as well as eschange
Cl these cffccts arc
second approximation,
we admit polarization
and concepts familiar
found m papers by hkWeeny
WF $o, and the modifications
and delocahzation.
and charge transfer occur beresulting therefrom
A correct link between complicated
to chemists can hence be established.
[9] (orthogonalized
WF in terms of the non-ortho-
groups which, at most, are Involved m each
the analysis should make clear which covalent bonding
tween the ofornic orbirals in the one-configuration wavefunctions
as well as to intramolecular
(WF) of
(Cl) expanded
to second order.
The aim of this letter is to investigate gonal valence-bond
wavefunction Interaction
which are closer to the full MO ones m the same basis than the simple 90 approach
This might be expected, since it is known [51 that a MO
induction
configuration
case) and Raimondi
when, in the IV-electron
Reference to ab mitlo VB theory
et al. [IO]
(non-orthogonal
can be
~3s~).
2. Theory For a system having a closed-shell structure,
(u) and m - II antibonding into the atomic orbital
from a mimmal set of 111atomic orbitals WC construct
(u*) orbitals (IV = 2rr is the total number of electrons).
basis gives a total of 3” singlet (S = 0) VB structures
Expansion
II bonding
of the o + o* basis
for the one-configuration
I&, 2 X 3”-’
structures for each polarization component, and 4 X 3 +* structures for each deloc3lization component. It can be shown, however, that when a single excitation involving any two groups at a time occurs, the possible StrucIurcs of the remaining II - 2 groups remain unchanged. We can then focus attention on 3 given pair of two&ctron bonds which, without loss of generality, may be called A and B, and whose electrons can, in the one-configuration approxibond orbltals mation, be assumed to occupy with opposite spin the bond orbitals oA and oB *. The (unnormaiized) * tn the ase
of oA or oB or both descrkiig a lone p3ir or an inner shell it ISsuflinent lo dcletc the atomic orbUs which are missing and the corresponding Shter determinants. Some comments under tic remaining struc1urcs should Ihen be spproprulely changed
Cl 009~2614/82/0000-0000/S
07.75 0 1982 North-Holland
447
CHEMICAL
Volume 87, number 5
9 Aprd 1982
PHYSICSLETTERS
themselves will be constructed, in turn. by the in-phase linear combination of suitably directed hybrids, each belongmg to its own subspace [3], say (a + b), (c + d). In the Cl process we admit a variety of single excitations from O~,IQ IO (unnorm3lized) antibonding orbitals ui, u*B obrained respectively from the out-of-phase combinations (a - b), (c - d). As usual, a horizontal bar will be used to denote fl spin, a double vertical bar a normalized Slater determmant of order N, while we have chosen to represent by a single vertical bar unnormalized determinants of order N omitting, for short, the M - 4 spin orbitals which wdl be left unchanged during the excitation. Using elementary properties of determinants, we obtain upon expansion the following results. (i) Otle-collfigwah-011 WF $0, . ..II =
lb,q&&
IIac+ad+bc+bd
= {lacEI + ladkI+ + {1x&1
covhnt cor3lmt
t Ibc?iEl)
on A. on D
TE+ii&i;3~
IbcZil f IbdZEl) + (laGal+ t {la&iI
lomc on A. cov&nt on B
...I1 ladEI} + (Ibcal+
7 IaciiEl + Isd%il
.r [bdzal}
+
lbdkl)
IbcbEl + lbdbiil
coralcnt on A. loruc on U
lomc on A, ionic on B
The one-configuration WF tie IShence seen to contzm structures describing covalent bonding within each WIgroup plus iomc structures on the same group occurring with identical weight. This is in agreement with the elementary theory of valence [I I j .
polarized
(ii) Local polanzatiorl
WF on A,
__.I1+ IluAuBE;EB
IIo~o~~~,E,
...lI = Ilac+ad-bc-bd
= {lacfidl+ IadFcl}- {Ibckilt Ibdkl}
mnicon A.
(adsal -lb&I
. ..!I + (c~spm+$ spin) - Ibdml
--
={+_I-;I}+{+_+_++_J-,+_-++}. covhnt
+ lacZl+
ZE+Z&k+~
iomc on A. iomc on B
on B
There are now no purely covalent structures nor singly polar structures ionic on B. The ionic structures on A now have opposite phases, and electron migration wlthin the A bond can occur depending upon the srgn of the first-order perturbation coefficient. Since thus IS the most elementary way of mtroducmg a polarity into the A bond, the corresponding WF can be aptly described as the A-polarization function. WF of A onto B,
(iii) Delocalkatiotz
Ilu*u c 0 BBAB
..
II + IluAa,E;aB
= Il cc-dd+cd-dc
_ _- -_ ac+ad+bc+bd
. ..I1 ...I1 + (ol spin*
= IladZ;il+ lcd3al) - {lbcEal+ Id&I} =
r’- - +_ ++/_- +\_}_
mterbond covr+znl-ionic
448
(A*B-) structures
+
(lacEal
spm) t
Icdacl} - #d&l
+ Id&Jl}
Volume 87. number 5
CHEMICAL PHYSICS LETTERS
9 Apnl1982
In the expansion of the determinants in the second row above, the first two contrrbutions vamsh because of the Pauli pnnciple (two same-spin electrons are put into the same spatial orbital). The resulting WF descrtbes oneelcctron charge transfer from A to B (A+B-) plus additional covalent bonding between orbitals nor directly bonded in &u. Unhke the former [ 121, the latter effect does not seem to have been emphasized before. These charge-transfer states (CTS) are particularly important for descrtbing the smallsecond-orderinduction which occurs when the two bonds are rather far apart. In fact, it IS now well established 113-161 that the introductron of CTS IS a convenient way to obnare the slow convergence in long range of the multipolar expansion of the induction energv for neutral systems, allowing, even for small bases, for suitable modifrcattons in the tail of the wavefunction of A-in the mgron near B, where the B-induction potential @( 1) has its greatest value. . ..]I + The same IS true, “mutatis mutandis”, for the local polarization function on B, namely Ilo,u~ii,,i5u . ..II + Ilu,,oBOAZA . ..II. . ..]I. and for the delocalmtion WF of B onto A, IIuAu~CA‘oB IlUAU,a,L$
3. Discussion Of the 20 singlet (8 = 0) VB structures which are due for s four-electron system described by four orbrtals [17], all covalent, singly polar and doubly polar structures are singled out upon expansion aithe many-electron WF in the improved bond-orbital approach, escept for the covalent structure descrrbing a different allocation of the primary bonds in Go and the two doubly polar structures descrrbing all electrons as being simultaneously on A or B. The relative weight of the different structures is determined by the magnitude and sign of the perturbatton cocfficrents. The expansion of the polarrzation correction to the WF is seen not to introduce new structures besrdes those already contamed in ao, but simply to delete the covalent srructure for the group in question, leaving its two polar structures rv18 opposire sip. On the contrary, the delocalizatron correction to the WF originates structures lror contamcd in Go and descnbing clrorge trarlsJcr as well as covalent bonding between the two groups mvolved in the cucitatron. The picture resulting from the VE3analysis is at the same time simple and physically appealmg. It gtves a key to establish the connectton between our BO approach [l-4] and different interpretatne schemes [ILIE-201 proposed within the molecular-orbttal method. It is seen that polarization, delocabzation, and charge transfer are differeru concepts all expressingthe SUIWphysical effect, namely the induction mteractton existing between electron groups more or less localized in different parts of each molecule. The one-configuration BO WF, which already grves a good starting approximation for saturated molecules, at least in the case of non-orthogonal orbrtals, IS substantially tmproved by the Cl process which admtts singleexcitations to antibondmgu* orbit&. smce such induction effects and their exchange component can be taken into account IOsecond order. The advantageof the BO approach hcs m the fact that the starting WF, chosen on the basis of chemical intuition, can systematically be improved by dcfinite steps, each having a well-defined and easily analyzable physrcal meaning, m a way which IS strictly connected with the techniques used in the perturbation theory of the electronic structure of molecules.
Acknowledgement Financial support by the ltahan Natronal Research Councd (CNR) and by the Mmistry of Pubhc Education @PI) under grants ur Theoretical Chemistry is gratefully acknowledged.
References [ 1 I V. Magn=co and G.F. Musso.Cbcm. Phys. Letters 84 (1981) 575. 121 G.F. Musso wd V. Magnssco,J. Cbem. SOC. Fyi&y Trans. II 78 (1982), lo be pubhshed. 131 V. n1agna.m and G.F. Musso.J. Cbem. Phys. 60 (197-l) 3744.
439
Volume 57, number 5
(41 G F. Russo and V. Magxwo. J. Chem. whys. 60 (1974) 3754. IS 1 W. bfQS. in:The world of quantum chemistry. eds. R. Daudel and B. Pullmsn (Reidel, Dordfecht, f6l G Ch~ksinski and 8. Jcz~orskt, Mol. Phys 32 (1976) 81. 171 V. hfqyasco. hfol Phys. 37 (1979) 73. [Sj AJ. Sadlg, hfol Phys. 39 (1980) 1249.
[9] R. McWeeny.Proc. Roy. Sac. A223 (1954) 63,306.
[ IOJ hl Rsimondr,W. Campon and hf. liarplus, hlol. Phys. 34 (1977) 1483. [I l] CA. Co&on 2nd 1. Father. Phd. hlag. 30 (1949) 386. [ 121K- Eitwa and K hlorokuma, Intern. J. Quantum Chcm. 10 (1976) 325. 1131 C. CMasinski and B. Jcziorski, Mol. Phys. 27 (1974) 649. [l-l] V. hiagnxxo,C. FIgartand hf. Battezzatl. hfo1. Phys 34 (1977) 1201. [IS] hl. Battezzatt and V. M~gnssco,Chem.Phys. Letters67 (1979)471. 1161 V. Mgruscozmd G. Ronallo.Chcm Phys. Letters 79 (1981) 175. (171 V. hkgmco and G F. hlusso, 1. Cltem. whys. 47 (1967) 1723. [ 181 H. Ekmmo, Bull.C&em.Sot Japan 37 (1964) 1574. [ 191 TX. Brunck and r. Mxnhold, J. Am Chcm. Sot. 101 (1979) 1700. [20/ W H. ~~~3ngbo. H.B. Schtsgel and S. Wolfe, J. Am. Gem. Sot. 99 (1977) 1296.
450
9 Aprt! 1982
CHEMICAL PHYSICS LETTEIRS
1974) p_ 3 1.
CHEhlICAL PHYSICS LEllXRS
Volume 87. number 5
A VALENCE-BOND
ANALYSIS
Valerio
and Gian Franc0 MUSS0
MAGNASCO
OF THE IMPROVED
BOND-ORBITAL
WAVEFUNCTION
Isritulo di Chimim Indusniale dell’llnrrersit~. 16132 Genw, Ita!,
Received 25 Jsnuary 1982
An anzdysis ol the unproved bond-orbnal wwclunction m terms of localizrd singletVB structures reveals the bondmg and chug transfer occurring between the four electrons wluch arc uwolrcd in each excitation. The phywal nature oi the undcrlymg approGrn3tion can hence be dlscussed.
1. Introduction We recently
[I,?]
proposed an improved
to localized doubly
antlbondiig
occupied
(o*) orbitals
to second order. Prehminary [3,4].
theory.
approach by Including
polanzation
and delocahza-
Singlet coupled configurations
dercribmg
sit@ excnmons
are allowed to mteract with the one-configuratlon
localized bondmg (u) orbitals applications
ergy results and wavefunctions
(90)
bond-orbital
tion effects in the second order of perturbation
through a non-orthogonal
to intermolecular
[l]
[6-81
to an extent determined
seemingly approximated
excitation.
SCF WF does include
[7_] problems give cn-
Induction
by the stze of its basis set, and in our perturbative
(VB) structures
More precisely,
the nature of the resultmg many-electron
concerning
the pairs of two-electron
as well as eschange
Cl these cffccts arc
second approximation,
we admit polarization
and concepts familiar
found m papers by hkWeeny
WF $o, and the modifications
and delocahzation.
and charge transfer occur beresulting therefrom
A correct link between complicated
to chemists can hence be established.
[9] (orthogonalized
WF in terms of the non-ortho-
groups which, at most, are Involved m each
the analysis should make clear which covalent bonding
tween the ofornic orbirals in the one-configuration wavefunctions
as well as to intramolecular
(WF) of
(Cl) expanded
to second order.
The aim of this letter is to investigate gonal valence-bond
wavefunction Interaction
which are closer to the full MO ones m the same basis than the simple 90 approach
This might be expected, since it is known [51 that a MO
induction
configuration
case) and Raimondi
when, in the IV-electron
Reference to ab mitlo VB theory
et al. [IO]
(non-orthogonal
can be
~3s~).
2. Theory For a system having a closed-shell structure,
(u) and m - II antibonding into the atomic orbital
from a mimmal set of 111atomic orbitals WC construct
(u*) orbitals (IV = 2rr is the total number of electrons).
basis gives a total of 3” singlet (S = 0) VB structures
Expansion
II bonding
of the o + o* basis
for the one-configuration
I&, 2 X 3”-’
structures for each polarization component, and 4 X 3 +* structures for each deloc3lization component. It can be shown, however, that when a single excitation involving any two groups at a time occurs, the possible StrucIurcs of the remaining II - 2 groups remain unchanged. We can then focus attention on 3 given pair of two&ctron bonds which, without loss of generality, may be called A and B, and whose electrons can, in the one-configuration approxibond orbltals mation, be assumed to occupy with opposite spin the bond orbitals oA and oB *. The (unnormaiized) * tn the ase
of oA or oB or both descrkiig a lone p3ir or an inner shell it ISsuflinent lo dcletc the atomic orbUs which are missing and the corresponding Shter determinants. Some comments under tic remaining struc1urcs should Ihen be spproprulely changed
Cl 009~2614/82/0000-0000/S
07.75 0 1982 North-Holland
447
CHEMICAL
Volume 87, number 5
9 Aprd 1982
PHYSICSLETTERS
themselves will be constructed, in turn. by the in-phase linear combination of suitably directed hybrids, each belongmg to its own subspace [3], say (a + b), (c + d). In the Cl process we admit a variety of single excitations from O~,IQ IO (unnorm3lized) antibonding orbitals ui, u*B obrained respectively from the out-of-phase combinations (a - b), (c - d). As usual, a horizontal bar will be used to denote fl spin, a double vertical bar a normalized Slater determmant of order N, while we have chosen to represent by a single vertical bar unnormalized determinants of order N omitting, for short, the M - 4 spin orbitals which wdl be left unchanged during the excitation. Using elementary properties of determinants, we obtain upon expansion the following results. (i) Otle-collfigwah-011 WF $0, . ..II =
lb,q&&
IIac+ad+bc+bd
= {lacEI + ladkI+ + {1x&1
covhnt cor3lmt
t Ibc?iEl)
on A. on D
TE+ii&i;3~
IbcZil f IbdZEl) + (laGal+ t {la&iI
lomc on A. cov&nt on B
...I1 ladEI} + (Ibcal+
7 IaciiEl + Isd%il
.r [bdzal}
+
lbdkl)
IbcbEl + lbdbiil
coralcnt on A. loruc on U
lomc on A, ionic on B
The one-configuration WF tie IShence seen to contzm structures describing covalent bonding within each WIgroup plus iomc structures on the same group occurring with identical weight. This is in agreement with the elementary theory of valence [I I j .
polarized
(ii) Local polanzatiorl
WF on A,
__.I1+ IluAuBE;EB
IIo~o~~~,E,
...lI = Ilac+ad-bc-bd
= {lacfidl+ IadFcl}- {Ibckilt Ibdkl}
mnicon A.
(adsal -lb&I
. ..!I + (c~spm+$ spin) - Ibdml
--
={+_I-;I}+{+_+_++_J-,+_-++}. covhnt
+ lacZl+
ZE+Z&k+~
iomc on A. iomc on B
on B
There are now no purely covalent structures nor singly polar structures ionic on B. The ionic structures on A now have opposite phases, and electron migration wlthin the A bond can occur depending upon the srgn of the first-order perturbation coefficient. Since thus IS the most elementary way of mtroducmg a polarity into the A bond, the corresponding WF can be aptly described as the A-polarization function. WF of A onto B,
(iii) Delocalkatiotz
Ilu*u c 0 BBAB
..
II + IluAa,E;aB
= Il cc-dd+cd-dc
_ _- -_ ac+ad+bc+bd
. ..I1 ...I1 + (ol spin*
= IladZ;il+ lcd3al) - {lbcEal+ Id&I} =
r’- - +_ ++/_- +\_}_
mterbond covr+znl-ionic
448
(A*B-) structures
+
(lacEal
spm) t
Icdacl} - #d&l
+ Id&Jl}
Volume 87. number 5
CHEMICAL PHYSICS LETTERS
9 Apnl1982
In the expansion of the determinants in the second row above, the first two contrrbutions vamsh because of the Pauli pnnciple (two same-spin electrons are put into the same spatial orbital). The resulting WF descrtbes oneelcctron charge transfer from A to B (A+B-) plus additional covalent bonding between orbitals nor directly bonded in &u. Unhke the former [ 121, the latter effect does not seem to have been emphasized before. These charge-transfer states (CTS) are particularly important for descrtbing the smallsecond-orderinduction which occurs when the two bonds are rather far apart. In fact, it IS now well established 113-161 that the introductron of CTS IS a convenient way to obnare the slow convergence in long range of the multipolar expansion of the induction energv for neutral systems, allowing, even for small bases, for suitable modifrcattons in the tail of the wavefunction of A-in the mgron near B, where the B-induction potential @( 1) has its greatest value. . ..]I + The same IS true, “mutatis mutandis”, for the local polarization function on B, namely Ilo,u~ii,,i5u . ..II + Ilu,,oBOAZA . ..II. . ..]I. and for the delocalmtion WF of B onto A, IIuAu~CA‘oB IlUAU,a,L$
3. Discussion Of the 20 singlet (8 = 0) VB structures which are due for s four-electron system described by four orbrtals [17], all covalent, singly polar and doubly polar structures are singled out upon expansion aithe many-electron WF in the improved bond-orbital approach, escept for the covalent structure descrrbing a different allocation of the primary bonds in Go and the two doubly polar structures descrrbing all electrons as being simultaneously on A or B. The relative weight of the different structures is determined by the magnitude and sign of the perturbatton cocfficrents. The expansion of the polarrzation correction to the WF is seen not to introduce new structures besrdes those already contamed in ao, but simply to delete the covalent srructure for the group in question, leaving its two polar structures rv18 opposire sip. On the contrary, the delocalizatron correction to the WF originates structures lror contamcd in Go and descnbing clrorge trarlsJcr as well as covalent bonding between the two groups mvolved in the cucitatron. The picture resulting from the VE3analysis is at the same time simple and physically appealmg. It gtves a key to establish the connectton between our BO approach [l-4] and different interpretatne schemes [ILIE-201 proposed within the molecular-orbttal method. It is seen that polarization, delocabzation, and charge transfer are differeru concepts all expressingthe SUIWphysical effect, namely the induction mteractton existing between electron groups more or less localized in different parts of each molecule. The one-configuration BO WF, which already grves a good starting approximation for saturated molecules, at least in the case of non-orthogonal orbrtals, IS substantially tmproved by the Cl process which admtts singleexcitations to antibondmgu* orbit&. smce such induction effects and their exchange component can be taken into account IOsecond order. The advantageof the BO approach hcs m the fact that the starting WF, chosen on the basis of chemical intuition, can systematically be improved by dcfinite steps, each having a well-defined and easily analyzable physrcal meaning, m a way which IS strictly connected with the techniques used in the perturbation theory of the electronic structure of molecules.
Acknowledgement Financial support by the ltahan Natronal Research Councd (CNR) and by the Mmistry of Pubhc Education @PI) under grants ur Theoretical Chemistry is gratefully acknowledged.
References [ 1 I V. Magn=co and G.F. Musso.Cbcm. Phys. Letters 84 (1981) 575. 121 G.F. Musso wd V. Magnssco,J. Cbem. SOC. Fyi&y Trans. II 78 (1982), lo be pubhshed. 131 V. n1agna.m and G.F. Musso.J. Cbem. Phys. 60 (197-l) 3744.
439
Volume 57, number 5
(41 G F. Russo and V. Magxwo. J. Chem. whys. 60 (1974) 3754. IS 1 W. bfQS. in:The world of quantum chemistry. eds. R. Daudel and B. Pullmsn (Reidel, Dordfecht, f6l G Ch~ksinski and 8. Jcz~orskt, Mol. Phys 32 (1976) 81. 171 V. hfqyasco. hfol Phys. 37 (1979) 73. [Sj AJ. Sadlg, hfol Phys. 39 (1980) 1249.
[9] R. McWeeny.Proc. Roy. Sac. A223 (1954) 63,306.
[ IOJ hl Rsimondr,W. Campon and hf. liarplus, hlol. Phys. 34 (1977) 1483. [I l] CA. Co&on 2nd 1. Father. Phd. hlag. 30 (1949) 386. [ 121K- Eitwa and K hlorokuma, Intern. J. Quantum Chcm. 10 (1976) 325. 1131 C. CMasinski and B. Jcziorski, Mol. Phys. 27 (1974) 649. [l-l] V. hiagnxxo,C. FIgartand hf. Battezzatl. hfo1. Phys 34 (1977) 1201. [IS] hl. Battezzatt and V. M~gnssco,Chem.Phys. Letters67 (1979)471. 1161 V. Mgruscozmd G. Ronallo.Chcm Phys. Letters 79 (1981) 175. (171 V. hkgmco and G F. hlusso, 1. Cltem. whys. 47 (1967) 1723. [ 181 H. Ekmmo, Bull.C&em.Sot Japan 37 (1964) 1574. [ 191 TX. Brunck and r. Mxnhold, J. Am Chcm. Sot. 101 (1979) 1700. [20/ W H. ~~~3ngbo. H.B. Schtsgel and S. Wolfe, J. Am. Gem. Sot. 99 (1977) 1296.
450
9 Aprt! 1982
CHEMICAL PHYSICS LETTEIRS
1974) p_ 3 1.