elementary finite order perturbation theory for vertical ionization energies

Elementary finite order perturbation theory for vertical ionization energiesGregory Born, Henry A. Kurtz, and Yngve Öhrn Citation: The Journal of Chemical Physics 68, 74 (1978); doi: 10.1063/1.435475 View online: http://dx.doi.org/10.1063/1.435475 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/68/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in WKB energy quantization and first-order perturbation theory Am. J. Phys. 65, 320 (1997); 10.1119/1.18749 A test of partial third order electron propagator theory: Vertical ionization energies of azabenzenes J. Chem. Phys. 105, 2762 (1996); 10.1063/1.472138 Large order dimensional perturbation theory for complex energy eigenvalues J. Chem. Phys. 99, 7739 (1993); 10.1063/1.465703 Vertical and adiabatic ionization energies of NH− 4 isomers via electron propagator theory and many bodyperturbation theory calculations with large basis sets J. Chem. Phys. 87, 3557 (1987); 10.1063/1.453000 Contribution of triple substitutions to the electron correlation energy in fourth order perturbation theory J. Chem. Phys. 72, 4244 (1980); 10.1063/1.439657

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Elementary finite order perturbation theory for vertical ionization energiesa)

Gregory Born, Henry A. Kurtz, and Yngve Ohm

The Quantum Theory Project, Department of Chemistry, The University of Florida, Gainesville, Florida 3261 1 (Received 27 June 1977)

The analysis of ionization energies in Rayleigh-Schrodinger perturbation theory and in proragator theory, previously known separately through third order in electron interaction, are compared in detail using elementary algebraic methods and their equivalence is explicitly shown. Relaxation terms are identified as the Ll. ESCF contributions and appropriate rules are described for their construction from the electron propagator diagrams of arbitrary order. Correlation terms are obtained separately. The transition operator method is analyzed and found to differ from to ESCF in third order, contrary to earlier claims.

I. INTRODUCTION

Propagator theory1 as a direct method for calculating ionization ,energies has been used for a large number of atomic and molecular systems with very good results. Numerous applications have commented upon the limitations of the approximation referred to as Koopmans' theorem,2 and some detailed studies.have been made of the errors made in this assumption. Pickup and Goscinski3 have given a detailed analysis of the error terms through second order in perturbation theory and they have stressed the propagator equations4 as the best direct method to obtain ionization energies. The more recent work of Cederbaum,5 Purvis and Ohrn, 6,7 and Simons et al. 8 indicated clearly that when a M,sUerPlesset9 partitioning of the Hamiltonian is used, terms through third order in perturbation theory are necessary in order to produce a predictive theory.

Some of the more readily obtainable correction terms are referred to as relaxation corrections3 and can be calculated by a AEsCF procedure, 10 i. e., performing separate Hartree-Fock calculations on the N-electron ground state and the desired (N -I)-electron state and subtracting the resulting total energies so as to obtain a theoretical ionization energy. This indirect manner of obtaining electron binding energies has of course the usual disadvantage of loss of accuracy in subtracting two nearly equal, large numbers. A direct method for calculation of relaxation effects have also been devised and given the name transition operator method. 11

In this paper we analyze the calculation of vertical ionization energies using Rayleigh-Schrodinger perturbation theory through third order and the M,sller-Plesset

I

-I(ASCF) =EHF(N) -E'fiF(N -1)

partitioning of the Hamiltonian. We study in this manner the AEsCF ionization energies in Sec. II, and in Sec. III we analyze the transition operator method. We find contrary to earlier claimsll that the transition operator method does not give the AEsCF energies correct through third order.

In Sec. IV we give a brief summary of the electron propagator results for ionization energies through third order in electron interaction. We devise diagrammatic rules for obtaining the relaxation terms from the propagator diagrams, and in Sec. V we analyze the correlation corrections, and show the expected result that the indirect calculation of differences between stationary state energies of the N-electron ground state and the appropriate state of the (N - I)-electron system yields the same result as the direct calculation when approximated to third order in Rayleigh-Schrodinger perturbation theory. We work with algebraic methods and identify the resulting terms with propagator diagrams following the rules of Goldstone and Hugenholtz as modified by Brandow. 12

II. ANALYSIS OF .::lEscF

The analysis requires expreSSions for the total energies for the N-electron ground state electron configuration as well as a configuration of the (N - I)-electron system differing in only one spin orbital, which we denote by x. This means that at least one of the systems is of the open shell type and we choose to work entirely with spin orbitals and a spin unrestricted Hartree-Fock scheme. The following expreSSion is then obtained for the negative of the ionization energy:

=[L:E:a--21 L:L: (abl[ab)]- L:Ea-~ L: (ablilib)] a ab ah ab~:r

=E:x+ L: (E. - Ea) - L: (ax I lax) ..: ~ L: [(abllab) -(abllab)] , a:;!x a'l-r; al#r

(1)

where the labels with a tilde denote the orbital energies and spin orbitals of the (N -I)-electron system.

a)This research was sponsored by a grant from the Air Force Office of Scientific Research.

I Throughout the paper we will use the notation a, b, c, ... for occupied spin orbitals, p, q, r, ... for unoccupied, and i, j, k, ... for unspecified spin orbitals. The antisymmetric two-electron integrals are

74 J. Chern. Phys. 68( 1), 1 Jan. 1978 0021-9606/78/6801-0074$01.00 © 1978 American Institute of Physics


On: Tue, 02 Dec 2014 07:01:26

Born, Kurtz, and Ohrn: Vertical ionization energies 75

(abllab)= f a"'{I)b*(2)ri~ (1-P12)a{l)b(2)d{l)d{2) , (2)

with P 12 a permutational transposition operator.

When the N-electron ground state Hartree-Fockproblem is solved we have a set of spin orbitals and a Fock operator

F(N) = h + L (a II a) , (3) a

with

(a II a) = f d(2)a* (2) ri~ (I - P 1Z) a(2) (4)

defining the unperturbed problem. The Fock operator for the (N - I)-electron problem can then be expressed as

F,,{N-l)=h+ L (alia) a~"

=F{N)-<xllx)+ L [(alla)-(alla)] , (5) a~

- I{ASCF) = E:" - 2: (E:!l) H!Z) H!3» - L: (ax II ax) at-x ai-x

where V = V1 + Vz, with

V1=-<xllx) , and Vz=L: [(alla)-(alla)] (6) a~"

is to be treated as aperturbation. This kind of treatment in the same notation has been discussed by Goscinski and co-workers. 11

We employ Rayleigh-SchrOdinger perturbation theory to calculate the spin orbitals and energies through a certain order

(7)

and since the perturbation V Z depends on the perturbed spin orbitals, the analysis is done iteratively as demonstrated below, and we count orders in terms of the electron interaction. Through third order we get straightforwardly from Eq. (1):

+ LL [(a(1)bllab)+(ab lla(1)b)+(a(Z)bllab)+(ablla(Z)b)+(a(l)blla(1)b)] ab~"

(8)

Noting that

~ E!l) = ~ (al Via) =-~ (axil ax) + ~~ [(abllab) -(abll ab)]

=- L: (axllax)+ L:)' [(ab(1)llab)+(abllab(1»+(ab(Z)llab)+(abllab(2»+(abU)1lab U»] .~ ab'l;-'

(9)

we can cancel several terms and obtain

-I{ASCF)=E:,,-:E (E:!ZlH~3»+! 2:L: «a(1)b(1lllab)+(abllaU)bU»+(a(1)bllab<1l)+(abUllla(1)b». (10) a~ 2 ab~

Let us consider one example in some detail

SUbstitute V1 + Vz for V and keep only terms which will contribute through third order

and use the expression

b(ll=_ L 1 (lIVlb) m (E:,-E: b)

This yields the following expression:

J. Chem. Phys., Vol. 68, No.1, 1 January 1978

(11)

(12}

(13)


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76 Born, Kurtz, and Ohrn: Vertical ionization energies

LE~2)=-LL l(axllkx)12 a~x a~ k~a (Ek - Ea)

+ LLLL[(axllkX)(klllab)(bxlllX) a~ b~x k~a Ub (Ek - Ea)(E, - Eb)

(ax I I kx) (kb I I al)(lx I I bx) (all I kb)(bx I Ilx)(kx I I ax) (ab I I kl)(lx I I bx)(kx I I ax)] + + +

(E k - Ea)(E, - Eb) (Ek - Ea)(E, - Eb) (Ek - E.HE ,- Eb) (14)

Similarly, we obtain for the other terms of Eq. (10):

L E~3) = - L L L: (lxll kx)(kxll ax)(axlllx) + L L (lxll ax) (ax I I ax)(axlllx) a~x a~ '~a k~a (E,-Ea)(Ek-E a) a~x '~a (E,-E a)2

(15)

and

=! L L L L [(klll ab)(axll kx)(bxlllx) 2 a~x b~x k~a '~b (E k - Ea)(E, - Eb)

(ab I I kl)(kx I I ax)(lx I I bx) (kb I I al)(ax I I kx)(lx I I bx) (all I kb)(bx I Ilx)(kx I I ax)] + + +

(Ek-EaHE,-E b) (Ek-Ea)(E,-E: b) (Ek-Ea)(E,-E b) (16)

The summations over all spin orbitals in Eqs. (14)-(16) will now be divided up into sums over occupied and sums over unoccupied spin orbitals in order to make contact with diagrammatic expansions. Considering the first term in Eq. (14) we see that the part with k occupied is zero, so we can write for the nonzero part

I (ax I Ipx) 12 (Ep-Ea)

where we have removed the restriction on the index a, since the term for a =x vanishes. The next step is to combine the last four terms in Eq. (14) and Eq. (16)

_ ! L L L L [(axl I kx)(klll ab)(bxlllx) 2 a b k~a m (Ek-Ea)(E,-E b)

(ax I I kx)(kb I I al)(lx I I bx) (ab I I kl)(lx I I bx)(kx I I ax) (all I kb)(bx I Ilx)(kx I I ax)] + + +

(Ek-Ea)(E,-E: b) (Ek-Ea)(E,-E b) (Ek-Ea)(E,-E b) (17)

We can again remove the restriction on the summation indices a and b, since the terms for a =x, and b =x are zero. When we now restrict the summations over k and l to occupied spin orbitals it is easily seen that this entire expression vanishes. We can explicitly put k = e, and l = d in Eq. (17), and exchange the labels b and d in the first term, which shows it to be the negative of the second term. Similarly, exchanging labels b and d in the third term cancels it against the fourth.

Next we consider the case when k is summed over occupied spin orbitals and lover unoccupied and viee versa. Again the entire expression vanishes. This can be seen by putting, say k =e and l =d, and interchange the labels a and e in the first term of Eq. (17), which shows that it cancels the fourth term. Similarly, exchanging the labels a and e in the second term cancels it against the third. When both k and l are summed over unoccupied spin orbitals we obtain a nonzero contribution

- LL L L(pxllax)(aqllpb)(bxllqx) a b P. (Ep-Ea)(E.-Eb)

where we have put k =p and l =q, and recognized that the second term in Eq. (17) then can be made identical to the fourth term by interchanging the labels a and b, and also interchanging p and q.

Finally, we attend to the two terms of Eq. (15). When both k and l are summed over occupied spin orbitals we can combine the two terms to the expression

_ LLL (ex I I bx)(bxll ax) (ax I I ex) , a~b~e (Eb - EaHEe - Ea)

(19)

where we have put k = b and l = e. This sum is zero, which can readily be seen by multiplying numerator and denominator with (Eb - Ee) and separate into two terms

_ E (ex I I bx)(bx I I ax) (ax I I ex) ] . e ~b-EJ~e-EJ~b-EJ

Cyclic relabeling a- e- b- a in the second term explicitly demonstrates that the two terms cancel.

When both k and l run over unoccupied spin orbitals we put k =p, l =q, and obtain from the first term of Eq. (15)

J. Chern. Phys., Vol. 68, No.1, 1 January 1978


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Born, Kurtz, and Ohm: Vertical ionization energies 77

- LL L (qxllpx)<pxllax)(axllqx) o p q (Eo-E)(Eo-Eq)

(20)

When k is summed over occupied and lover unoccupied spin orbitals, and vice versa, we obtain from the first term in Eq. (15)

- L LL [(CXIIPX)(pxllax)(axIICX) o eta p (€" - EoH Ep - €,,)

(pxllax)(axll cx)(cxIIPX)] - (€"-E""HEp-Ec ) •

(21)

The second term in Eq. (21) is obtained by putting k = c, 1 =p, and interchanging a and c. The two terms are then brought together by the identity

A ( 1 1) A p-a - p-c =- (p-a)(p-c) (22)

c -a

obtaining

(23)

where the terms with a = c are those from the second term in Eq. (15) with 1 =p summed over unoccupied spin orbitals. We have now obtained all nonzero terms contributing to -I(ASCF) in Eq. (10), but we will do one further rewriting of the last two sums in Eq. (18), which will make it easier to later connect with diagrammatic expansions. The first of those sums can be expressed as

(24)

where we have used the identity (22) with p = 0, c = Eo - Ep , and a = Eq - Eb • The second term on the right hand side of Eq. (24) can now be made explicitly identical to the first by interchanging labels a and b, as well as labels p and q. This results in the following expressions for the right hand side:

_ LLLL (axllpx)(pqllab)(bxllqx) o b P q (Eb - Eq) (E"" + Eb - Eq - Ep)

Similar treatment of the last sum in expression (18) results in

(25)

In summary, we list all contributions to - I(ASCF) through third order, i. e., all relaxation terms through third order contributing to the ionization energy associated with removing an electron from spin orbital x:

-I(ASCF)=Ex

- LL l(axllpx)12

- LLLL (axllpx)(pbllaq)(qxllbx) a p Ea - Ep 0 b P q (Eo - ~H Eb - Eq)

_ LLLL (axllpx)(pqllab)(bxllqx) _ LLLL (abllpq)(qxllbx)(pxllax) a b P q (Eb-Eq)(Ea+Eb-Ep-Eq) a b p q (Eb-Eq)(E",,+E:t,-Ep-Eq)

+ LL L (qxllpx)(pxllax)(axllqx) _ LL L (cxllpx)(pxllax)(axll cx) . a P q (E"" - ~)( E"" - Eq) a c P (E"" - Ep)( €" - Ep}

(26)

III. ANALYSIS OF THE TRANSITION OPERATOR METHOD

The transition operator methodll (TOM) calculates the ionization energy as the eigenvalue Ex to the "transition operator"

Fx=h+L (alla)+~<Xllx), a~"

with the spin orbitals j satisfying the equation

F"j =EJj .

(27)

(28)

One could proceed as in the previous section and write

F,,=F(N)+V, (29)

introducing a perturbation treatment starting from the N-electron Hartree-Fock ground state. This will not

give complete agreement of third order terms between - I(ASCF) and E" as observed by Goscinski et al. 11 A proof has been given by the same authors that when the TOM spin orbitals and corresponding energies are chosen as the reference for a Rayleigh-Schrodinger perturbation theory treatment of the total energy of the N-electron Hartree-Fock ground state and of the total HF energy of the desired electron configuration of the (N - 1)electron system, then the expression for -I(ASCF) as given in Sec. II agrees with E". This means that the transition orbital energy gives all relaxation terms correct through third order in electron interaction.

In this section we reinvestigate this proof. We denote with a tilde quantities belonging to the N -electron [or (N - I)-electron] Hartree-Fock state, and we consider the expansions



On: Tue, 02 Dec 2014 07:01:26


(30)

The unperturbed problem is given by Eq. (28), and we have

E(N)= L Eo- ~ L) (~b\\~b) o 0 ~

F(N)=F~+t(x\\x)+ L [(ii\\ii)-(a\\a)] (31) o

defining the perturbation, with F(N) given by Eq. (3), but now with spin orbitals labeled by a tilde.

The total Hartree-Fock energy for the N-electron ground state can be expressed as

= L (Eo+E~1)+E!2)+E!3»)_ ~ LL (ab\\ab)- LL (ab O)\\ab)+(ab\\ab(1»+(ab(2)\\ab) a a b a b

+ (ab \\ ab (2» + (abO) \\ abO») - ! L L (a(1)b (1) \\ ab) + (aO)b \\ ab (1» + complex conjugate) , 2 0 b

(32)

which is correct through third order. As in the previous section this can be simplified by considering explicitly the following expression:

L E~l)= L (a\V\a)=t(ax\\ax)+ LL [(ab\\ab)-(ab\\ab)] a a a b

=t(ax\\ax)+ LL [(ab(1)\\ab)+(ab\\ab O»+(ab(2)\\ab)+(ab\\ab(2»+(ab(1)\\ab(1»] , (33) o b

also correct through third order. We can then write the total energy as

For the desired state of the (N -I)-electron system we have the Fock operator

F(N -1) =h+ L (a\\a) , (35) o~~

where we have also denoted the spin orbitals with a tilde although they of course are different from those of the N-electron ground state. Again resorting to RayleighSchrodinger perturbation theory starting from the transition operator we obtain

F(N-l)=F~-~(x\\x)+ L (a\\a)-(a\\a»). (36) o¢,.

Identical treatment to that which led to Eq. (34) will now give

+ complex conjugate) . (37)

When we now consider the difference E(N) -E(N -1) we see that the leading term is the transition operator eigenvalue E~, and also that all first order terms explicitly cancel. Further cancellations are possible to recognize if we note that the leading term in the perturbation pertinent to Eq. (34) is ~ (x I I x), while the corresponding term for Eq. (37) is -~(xllx), Le., they

(34)

are equal in magnitude but opposite in sign. It is also important to note that the restriction on indices a and b in the sums of Eq. (37) is of no consequence, since the terms with a =x and b = x vanish, as was shown in Sec. II. Referring back to Eqs. (11)-(13) we see that in the expression for E!2) the leading term in the perturbation enters an even number of times in each term, and thus its sign has no effect. All these considerations lead to the conclusion that the terms involving E!2) cancel through third order when the difference E (N) - E(N - 1) is calculated. Similarly, we have exact cancellation through third order of the last four terms of Eq. (34) against the corresponding terms of Eq. (37).

This leaves only the terms involving E!3). For the terms in the expression of E(N) we have

(ll VI k)(kl VI a)(al VI!) (El - Eo) (Ek - Eo)

_ L L (ll V I a)(al VI a)(~1 Vll) ,

o ~o ~/-EJ (38)

with V =~ (xlix). The terms in the expression of E(N -1) look the same, but have V = - ~ (xlix). This means that when we take the difference these terms in the two expressions will not cancel, but add to the total result for the ionization energy

- I(ASCF) =E(N) - E(N - 1) = E~

+! LL L (lxll kx)(kxll ax)(axlllx) 4 0 1~0 ~o (E 1 - Eo)(Ek - Ea)

-! LL: (lxllax)(axllaxj(axlllx) (39) 4 0 1*0 (E 1 - Eo)



On: Tue, 02 Dec 2014 07:01:26


FIG. 1. The relaxation error (r) and the correlation error (e) are displayed as vertical lines at the corresponding ionization energies for the water molecule. The numbers are estimated from the experimental results as quoted in Ref. 15 and from f:l.ESCF results reported in Ref. 11.

It should be noted that while the expression for the ionization energy in Eq. (26) is given in terms of the Hartree-Fock spin orbitals and corresponding orbital energies of the N-electron ground state, the expression in Eq. (39) is given in terms of the transition spin orbitals and corresponding orbital energies. It is, howver, clear from this analysis that the transition orbital energy E: .. is not exactly equal to -I(f:l.SCF) through third order in electron interaction. Numerical calculations for core ionizations show the third order terms in Eq. (39) to be small, and the transition operator method is a very convenient way to account for most of the relaxation effects in an ionization energy.

Detailed calculations have shown that correlation effects are also important, and in Fig. 1 we show the contributions to the ionization energies from relaxation terms [-I(f:l.SCF) or transition orbital energy], and further correlation terms. It is interesting to note how the relative importance of these two kinds of contributions vary as one progresses from the core region to the outer valence electrons. It should also be recognized that the relative importance of these two kinds of effects is changed when one goes from a delocalized to a localized set of spin orbitals. This has been discussed by several authors. 13 The large relaxation contribution to the core ionization energies is of course well known. The correlation contribution being relatively important and positive in the inner valence region is a trend which we can observe in many systems, as well as it going negative in the outer valence region thereby counteracting the relaxation error and often making Koopmans' theorem give excellent results in this region for electron binding energies.

IV. THE ElECTRON PROPAGATOR

The difference in total energy of the N-electron system ground state and a particular state of the (N - 1)electron system can also be calculated directly beyond the f:l.ESCF approximation. This is most conveniently done using the electron propagator. 5,6 In practical calculations one uses the Hartree-Fock spin orbital basis of the N -electron ground state, and the M,dller-Plesset partitioning of the many-electron Hamiltonian

(40)

F 1(N)=h(1)+ L(a!!a) (41) a

is the Fock operator for electron lable 1, as defined in Eqs. (3) and (4).

The Dyson equation14 for the electron propagator matrix G (E) can then be written as6

(42)

where Go(E) is the Fock operator resolvent in the spin orbital basis.

Since each element G u(E) of the electron propagator matrix has Simple poles at values of the energy parameter, E =E(N) -E .. (N - 1), this means that ionization energies should satisfy the equation

(43)

where we have used the fact that Go(E) is diagonal in the Hartree-Fockspin orbital basis. The self-energy matrix ~(E) has been studied in many different approximations5

-7 and it is generally recognized that one has to

calculate it at least through third order in electron interaction to get good results. Theoretical results have been obtained this way, which compare well with results from photoelectron measurements of principal ionization energies as well as satellite structures in the spectra. Rayleigh-Schrodinger perturbation theory for the principal ionization energies is recovered through third order if we put

(44)

i. e., take the diagonal element of Eq. (43) and evaluate the diagonal element of the self-energy matrix for E =E: .. ,

the particular orbital energy that belongs to the ionized electron. To extend this analysis beyond third order, one must include both reducible and irreducible selfenergy contributions. 16

Equation (43) has been studied through second order in electron interaction by Pickup and Goscinski,3 and computationally through third order by several authors.5,B

We list below, through third order in electron interaction, all contributions to ~II(E) including their diagrammatic representation using the convention of antisymmetric vertices as discussed by Brandow12

]--7\ ___ V

• ]] (45)



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(3) _! L:L:L:L:L: (iallpq>(pqllrs>(rsllja> ~IJ (E)- 4 a p 0 r • (E+E.-Er-E.)(E+Ea-Ep-E o) J:~ (A)

-! L:L:L:L:L: (i£llab)(abllcd>(cdlljPz I-TI (B) 4. bcdP (E +Ep -Ec - Ed)(E +Ep - Ea -E b)

-L:L:L:L:L: (ibllrq)(rall£W.£'fJ.11ja) I-V (C) a b p 0 r (E+E.-Ep-Eq)(E+Eb-Eo- Er)

+ L:L:L:L:L: (iq! ICb>(c£llaq)(ablljp) abc Po (E+Ep-E.-Eb)(E+Eo-Eb-Ec) J:il (D)

+! L:4=L:L L: (icllab)(abll£v(£qlljc) 4 a c p • (E+Ec-Ep-Eo){E.+Eb-Ep-Eo) :f~ (E)

+ ! Ll:l:LL (iall£fJ.)(MII bc)(bcllja) 4 abc p 0 (E+E.-Ep-Eq)(Eb+EC-Ep-Eq) =tr:D (F)

+ ~ LLLL:L (i£ I I abl(ab I I qr)(qr I I j£) i:il (G) • b p 0 r (E +Ep - Ea -Eb)(Ea+Eb - Eo - Er)

+! LLLLl: (ipllqr)(qrllab)(ablljp) 4 • b P 0 r (E+Ep-Ea-Eb)(Ea+Eb-Eo-Er) ~:il (H)

_ LLLLL (irllafJ.)(abll£r)(Nlljb) • b p 0 r (E+Eb-Ep-Eq)(Ea+Eb-Ep-E,.) :fD (I)

_ LLL LL (ibl l£rZ(MI I ab)(arlljfJ.) =tr:il (J) a b p 0 r (E + Eb - Ep - Er)(Ea + Eb - Ep - Eo)

_ LLLLl: (iqllac)(abll£v(£clljb) abc P 0 (E+Eq-Ea-Ec)(Ea+Eb-Ep-Eo) ~:~ (K)

_ L~Ll:L (ic11qb)(qpllac>(abllj£> a cPo (E+Ep-Ea-Eb){Ea+Ec-Ep-Eo) ii:D (L)

+! L LL L L (ir Iljp>(ab I I rv(pq I lab) 2 a b p • ,. (Ea+Eb-Ep-E.)(Ea+Eb-E.-Er) >-~::~ (M)

- ~ l:LLLL (ialljc)(cbl Il!.fl)(£ql lab) >-fD (N) (Ea +Eb -Ep - Eo){Eb +Ec - Ep -Eo) abe P •

+! L4=l:1:L (i£llja)(abllqr)(flrlll!.b) ~] (0) 2 a P." (Ea+Eb-E.-E,.)(Ea-Ej» >---



On: Tue, 02 Dec 2014 07:01:26


Since we are here dealing with propagator diagrams rather than total energy diagrams, we have external (horizontal) lines for the unspecified indices i and j. The sign factor of a diagram is only affected by the number of internal "hole lines" and the number of closed loops.

The terms displayed in Eq. (46) do not discriminate between relaxation and correlation contributions, and the question arises whether diagrammatic rules can be designed so that, say, the relaxation terms can be obtained from the propagator diagrams in Eq. (46). This is in fact possible, and the following rules should augment the Brandow rules12 so as to give the relaxation or tl.ESCF contributions to the propagator diagrams:

(i) Consider only "hole propagator diagrams," i. e. , diagrams with the uppermost external line an incoming line.

(ii) Cut one or more internal.hole lines, but never more than one of a pair of equivalent hole lines, and keep only those resulting diagrams that have one incoming and one outgOing or no external lines per interaction.

(iii) Rearrange the "primitive" diagrams resulting from rule (ii) to "standard" form with all external lines representing the same electron coordinate and ignore any sign factors obtained in the process.

(iv) Apply the usual Brandow rules to the standard diagrams obtained from rule (iii) with the additional sign factor (_l)c+a+o+l, where C is the number of hole lines cut, a is the "adjacency number": 1 for external lines on adjacent interaction lines, 2 for external lines on interaction lines once removed, etc. for the parent propagator diagram, and 0 is the number of interaction lines.

As an example consider diagram (D) in Eq. (46). First we cut one hole line to obtain the diagram

}---T ---1-

which is next rearranged according to rule (iii) to

(p)

(Q)

(R) (46)

r-Applying the Brandow rules with an additional sign factor of (_1)1+2+3 [rule (iv)] we obtain the following algebraic expression:

_ L:L:L:L: (xallxp)(pbllaq)(xqllxb) a b P q (Ea-Ep)(Eb-Eq)

which is the third relaxation term in Eq. (26).

We can also cut two hole lines to obtain the diagram

>--~ >-->---wnich corresponds to the sixth relaxation contribution in Eq. (26).

Similarly, the propagator diagram (B) of Eq. (46) corresponds to the seventh relaxation term of Eq. (26), while diagrams (K) and (L) correspond to the fifth and fourth

relaxation terms, respectively. Note that the numerical factor of the resulting relaxation diagrams is obtained directly by the Brandow rules, i. e., a factor t for each pair of equivalent (internal) hole lines. For instance, the second order hole propagator diagram expression has a factor t due to the pair of equivalent hole lines, while the relaxation diagram resulting from cutting one of the hole lines corresponds to the second term of Eq. (26) with the numerical factor 1.

By using the procedure described by Eq. (44), and the terms of Eq. (46), we should now be able to extract the correlation terms through third order in electron interaction. We proceed to do this in the next section.

V. CORRELATION CORRECTIONS TO IONIZATION ENERGIES

The total electronic energy of the N -electron system can also be treated by Rayleigh-Schrodinger perturbation theory, using M,6ller-Plesset8 partitioning, and the total energy through first order is the Hartree-Fock energy, which we have used in Sec. II. We write

(47)

and a similar expreSSion for the desired (N - l)-electron state. In Secs. II and III we dealt entirely with

(48)



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which we called the relaxation part of the ionization energy. In this section we analyze the correlation corrections E(2)(N) _E!2l(N -1) and E!3l(N) _E!3 l(N -1). Pickup and Goscinski16 have made this analysis through second order in electron interaction.

(49)

where we have explicitly separated out the terms corresponding to spin orbital x. In the (N - I)-electron system the spin orbital x is in the unoccupied set, and we write

+! L L L <.Xiii ~b){~bll pq) , (50) 2 atx blx p (Ea+Eb-Ep-Ex)

where in the first term there is an implicit restriction in that p and q do not run over x. This becomes important when we look for cancellations between the expressions for E!2l(N) and E!2l(N -1) and want p and q in two summations always to run over the same set.

The Hartree-Fock spin orbitals and orbital energies of the (N - I)-electron system are denoted by a tilde, and can now be expanded in terms of the corresponding N-electron quantities as was done in Sec. II, Eq. (7).

Through third order in electron interaction we obtain contributions both from the spin orbital corrections j(1l

and from the orbital energy corrections E~1). While the former enter in a straightforward manner, it is worthwhile to comment in particular about the contributions from the latter. Considering the first term in Eq. (50) we have

(51)

with E(abpq) =E~ll +E~ll - E~ll - E~l) and Ekll = (kl VI k), where V = V1 + V 2 as defined in Eq. (6). Only V 1 contributes through third order and from the second term of Eq. (51) we get a number of terms, a typical one being

- ~ L L LL (pqll ab){xallxa){abll pq) a blx P q (E.+E b _Ep_Eq)2

(52)

The first terms of Eqs. (49) and (50) cancel upon subtraction, and the remaining (second order) correlation corrections to the ionization energy are

E!2l(N) _E!2l(N -1) =! LLL (xallpq)(pqllxa) +! '} '} L (xpll ab)(abllxp) 2 a p q (E .. +Ea-Ep-Eq) 2 ~ nt p (Ex+Ep-Ea-Eb)

-!LLLLL 2 alx blx p q kIp

+~~~~~~

The third through sixth terms in Eq. (53) contain the restricted summations k* a or k* p. It is possible to remove these restrictions and simplify the result. This is done by first splitting the sum over k into occupied and unoccupied parts.

Secondly, combinE\ complex conjugate pairs which have denominators containing the factor (Er - Ep) or the factor (Eo - Eel. This is done using the identity in Eq. (22). Finally, the summation restriction will be removed by adding the appropriate term from Eq. (53) (seventh through tenth terms). We demonstrate this proced~re for the complex conjugate pair of the third term in Eq. (53) for the case k =r, i. e., summing only over unoccupied spin orbitals



On: Tue, 02 Dec 2014 07:01:26

Born, Kurtz, and Ohrn: Vertical ionization energies

Nothing that the missing terms in Eq. (54), where r= p, are given by the eighth term in Eq. (53) we can now add these terms and remove the restriction on the summation over r.

83

(54)

Carrying out this procedure for the other terms in Eq. (53) we obtain six terms which correpond to six of the propagator diagrams in Eq. (46). The expression just derived in Eq. (54) corresponds to propagator diagram (M) in Eq. (46). Similarly, we have the correspondence between diagram (N) and the fourth and seventh terms; diagram (0) and the fourth term; diagram (p) and the third term; diagram (Q) and the third term (complex conjugate); and diagram (R) and the fourth term (complex conjugate). This correspondence is of course not precise, since the summations over occupied spin orbitals e. g., in Eq. (54), still exclude spin orbital x. In addition to these terms we obtain a number of expressions which will cancel some of the terms obtained from E(3)(N) - E(3)(N - 1). These are the following terms:

We will now proceed to consider E(3 1(N) and E(31(N - 1) and show that we obtain terms, which cancel terms three and four of Eq. (55), and also the remaining propagator third order diagrams will be recognized.

In third order we have the three total energy terms

As earlier we write these terms for the N-electron system by separating out the parts, which have one or more of the occupied spin orbitals equal to x. This is done to directly recover the terms, which correspond to the remaining propagator third order diagrams. We have

-LLLLL <Xallpq)<pbllra)<rqllxb) a b p q r

(56) The third order corrections for the (N -I)-electron system are written similarly with the spin orbital label x impliCitly removed from the summations over unoccupied spin orbitals. The expansion of the (N - I)-electron spin orbitals and orbital energies in terms of the N-electron quantities is trivial, since only the lowest order contributes, and we obtain:



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Upon subtraction of third order correlation corrections for the Nand (N - I)-electron systems the first terms of Eqs. (56) and (57), respectively, cancel. Similarly, the sixth terms of these equations and the tenth term of Eq. (56) and the eighth term of Eq. (57) cancel pairwise. Because of the manner in which we wrote the sums, we can immediately identify the following "particle" diagrams of the electron propagator:

c J E

The terms corresponding to the other propagator diagrams are obtained by combination of several expressions. For instance, subtracting the seventh term of Eq. (57) from the first of Eq. (55) and adding to this result term seven of Eq. (26) results in

_! LLLLL (abllxp)(cdllab)(xpllcd) , 4 abc d p (Ex+Ep-Ea-Eb){E,,+Ep-Ec-Ed)

(58) where the corresponding propagator diagram from Eq. (46) is also displayed, and the procedure of Eq. (44) ties the two together. Similarly, subtracting term four of Eq. (57) from the second of Eq. (57) and adding the relaxation term three and seven of Eq. (26) yields term (D) of Eq. (46). Even more tedious algebra may be involved, and as a typical example we consider the reconstruction of the full terms (P) and (K) of Eq. {46}. We start with the restricted sum {third term} of Eq. (26) for k running over occupied spin orbitals

and the second term of Eq. (57), which now has changed sign after the subtraction E(3){N) - E(3)(N -1):

A

_ L L L LL --,--..'::(a~b I~I PUqlL.-')(CLP'::";c 1....:...:1 x:::..::b.L!)(x=ql-'-1 c..::1 a=c)_ a~" bk ct" " q (E,,+Eq-Ea-Ec){Ea+Eb-E,,-Eq)

(60)

By subtracting the following term from expression (59):

(61)

and adding it to the expression (60) after changing label c into b, a into c, and interchanging labels p and q we obtain

- L LLLL (abllpq)(pcllxb)(xqllac) ck a b P q (Ex+Eq-Ea-Ec){Ea+Eb-Ep-Eq)

{62}

The first term in expression {62} now exactly equals the perturbation contribution to the ionization energy which corresponds to the electron propagator diagram {P} of Eq. (46). Addition of the relaxation term five from Eq.



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(26) removes the remaining restriction in the summation for the second term of expression (62). The result is easily identified as corresponding to diagram (K) of Eq. (46).

VI. CONCLUSION

We have shown how the analysis of corrections to the Koopmans' theorem results can be carried through third order in Rayleigh-Schrodinger perturbation theory. Relaxation terms have been identified as the AEsCF contributions and rules have been given how to obtain them from electron propagator diagrams. Detailed elementary algebraic manipulations have been stressed, and it is seen that through third order it is possible in a relatively straightforward manner to deal with all contributions to the ionization energies. It is also quite clear that a diagrammatic systematization of terms from the equation of motion for the electron propagator is very useful and elegant. It has also been shown that, contrary to popular belief, the transition operator eigenvalues are not equal to the AEsCF result for ionization energies through third order. The third order "error" terms are small in most cases but they are nevertheless present.

ACKNOWLEDGMENT

We gratefully acknowledge many conversations with Dr. M. Hehenberger about the transition operator method.

lJ. Linderberg and Y. Ohrn, Propagators in Quantum Chemis-try (Academic, London, 1973).

2T . A. Koopmans, Physic a (The Hague) 1,104 (1933). 3B . T. Pickup and O. Goscinski, Mol. Phys. 26, 1013 (1973). 4For a recent review see Y. Ohrn, "Propagator Theory of

Atomic and Molecular Structure," in The New World of Quantum Chemistry, Proceedings of the Second International Congress of Quantum Chemistry, New Orleans, 1976, edited by B. Pullman and R. Parr (Reidel, Boston, 1976).

5L. S. Cederbaum, Theor. Chim. Acta 31, 239 (1973); J. Phys. B 8, 290 (1975).

6G. D. Purvis and Y. 6hrn, J. Chem. Phys. 62, 2045 (1975); 65, 917 (1976).

1 •. L. Tyner Redmon, G. D. Purvis, and Y. Ohrn, J. Chem. Phys. 63, 5011 (1975).

8J. Simons, Chem. Phys. Lett. 25, 122 (1974); J. Simons and W. D. Smith, J. Chem. Phys. 58, 4899 (1973); K. M. Griffing and J. Simons, J. Chem. Phys. 62, 535 (1975).

9C. MOller and M. S. Plesset, Phys. Rev. 46, 618 (1934). lOp. S. Bagus, Phys. Rev. 139, 619 (1965). 110. Goscinski, B. T. Pickup, and G. D. Purvis, Chem. Phys.

Lett. 22, 167 (1973); O. Goscinski, M. Hehenberger, B. Roos, and P. Siegbahn, Chem. Phys. Lett. 33, 427 (1975).

12B. Brandow, Rev. Mod. Phys. 39, 771 (1967). 13L. C. Snyder, J. Chem. Phys. 55, 95 (1971); P. S. Bagus

and H. F. Schaefer III, J. Chem. Phys. 56, 224 (1972); L. S. Cederbaum and W. Domcke, J. Chem. Phys. 66, 5084 (1977).

14See for example Gy. Csanak, H. S. Taylor, and R. Yaris, Adv. At. Mol. Phys. 7, 287 (1971).

15J. Almlof, University of Stockholm Institute of Physics Report 72-09 (1972).

16p. W. Langhoff and A. J. Hernandez, Chem. Phys. Lett. 49, 421 (1977).



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elementary finite order perturbation theory for vertical ionization energies

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