Inapplicability of the Hubbard model for the description of real strongly correlated electrons

PHYSICA ELSEVIER

Physica B 199&200 (1994)366-372

Inapplicability of the Hubbard model for the description of real strongly correlated electrons .I.E. Hirsch Department of Physics, University of California. San Diego, La Jolla. CA 92093-0319. USA

Abstract The single band Hubbard model assumes that two electrons in an atom will doubly-occupy the same atomic orbital that is singly occupied when a single electron is at the atom. This assumption is incorrect, due to the fact that the spacing of atomic energy levels is always smaller than the intra-atomic Coulomb repulsion between electrons. Relaxation of this assumption leads to an effective single band Hamiitonian where the mteratomic hopping amplitude varies with the atomic charge occupation. First principles calculations of the variation in the magnitude of hopping amplitudes are presented and its possible consequences for superconductivity are discussed.

1. introduction The single band Hubbard Hamiltonian [1,2]

H = - t ~" {c~ocj. + h.c.) + U ~ n i . ni~ <~J, i ¢r

11)

is widely used to describe electronic correlations in narrow energy bands. It describes the physical fact that two electrons of opposite spin will tend to avoid occupancy of the same atom due to the large Coulomb repulsion U that they would experience. The intra-atomic Coulomb repulsion U is typically of order 10 eV and the hopping rate between neighboring sites, t, is typically assumed to be a fraction of an eV. This Hamiltonian is fnndamentally flawed because it ignores an energy scale that is in between t and U: the spacing between atomic energy levels, which we will denote by ~:. Besides avoiding each other by occupying

different atoms, real electrons will also avoid each other when they are on the same atom, by developing intraatomic correlations; that is, they will occupy different rather than the same atomic orbital. This effect will become important when the "band" described by Eq. I l) is more than half full so that two electrons are forced to be at the same atom at times. Consider the simplest atomic system, a hydrogen ion. One electron occupies the atomic orbital

~l"r}

/ ")'3 "\ ! 2 = f *~) K C

Zr

(2'

( Z - II. Wl-:ere will a second electron of opposite spin £o? if it occupies the same orbital, the two-electron wave ftmction is ~trl.r21 .... ;~t3rl1~pt,lr:!.

0921-452694 $07.(X) ~': 1994 Elsevier ~ctcnce BV. AP rights reserved SSDI Ot~21-4526193~ [:.026t)- N

43~

367

J.E. Hirsch / Physica B 199&200 (1994) 366-372

the two electrons experience a repulsion

U =(lslsll/rllsls)=

dSrld~r2lq~l~lrlil ~

5

x I~pt.,(r~)l~ = ~ Z = 17 eV,

(4)

and the total atomic energy is E=

-2Z 2+IZ=

-10.20eV.

(5)

Instead, the second electron will realize that there are other atomic orbitals that it can occupy. The next highest, the 2s orbital

=

z,v(

I

-

zI 5 r

e-

(6)

has an energy e. = ~ Ry = 10.20 eV

(7)

above the Is orbital given by Eq. (2). Because r. is smaller than U it will be advantageous to partially occupy this orbital to reduce the intra-atomic Coulomb repulsion, as the repulsion energy between one electron in the Is and another in the 2s orbital is only V = (ls2sl I rlls2s) = 5.71 eV.

(8)

tion to the Is and 2s orbitals, it can be seen that the simple wave fuuctiol; ~iv~a by Eq. i'~) goc~ ahnos~ 90% of the way in correcting the purely Is wave function given by Eq. (3) so, that it is certainly adequate for qualitative purposes. Now there is no reason in principle why the hopping rate of electrons between neighboring atoms should be the same fo:" I s and 2s orbitals. Rather, one would expect different hopping rates for Is-Is, ls-2s and 2s-2s hoppings. The wave function given by Eq. 19) shows that in the doubly occupied atom one electron spends more than 50% of the time in the 2s orbital; hence, there is no reason why the hopping rate to or from such a doubly occupied atom should be the same as that to or from a singly occupied atom, where only the Is orbital is involved. The Hamiltonian equation Ilk with a single hopping rate t that is independent of the charge occupation of the atoms, lacks the ability to describe this physics. How does one incorporate this physics? One possible way is to generalize Eq. (1) to describe two orbitals per site, with all the possible intra- and inter-atomic single and two-particle couplings between the orbitals. Because of the separation of energy scales that usually occurs, however,

1121

t ~, r, ~ U ,

it appears possible to keep the simple physics of one effective orbital per site, provided that v,e allow the hopping amplitude to vary with the atomic charge occupation. The Hamiltonian equation {1) generalizes to,

The wave function .'i3, o

Ip(r~,r:) = 0.683 ~01,(rl )'p~,(r2) + 0.731 x cpl,(rl ) q)2,(rz ) + (P2,(r 1) Ol,(r: )

i

tTj = toll - th.-,,)(l - rb._,) + ta(n~.-~ + nj._~ - 0.0146 (p2,(r i ) q)2,{r2 ) -- 2hi. _~,tlj. _~) + t2ni.-anj..-a.

(13b)

(9) has energy E=

- 13.89eV.

(I0)

substantially low'or than that given by Eq. 15~ iit is the optimal wave function involving these t~o orbital.q [3]. in fact, the exact energy for two electrons in hydrogen 1 t f - 1 [4J is E = -- ~4.36eV.

ill)

which would be obtained bv allowing the two electrons to partially occup? all the olher atomic orbitals in addi-

The hopping amplitude is now to, tl or t 2. depending on whethm :here are 0, 1 or 2 other electrons (besides the one that is hopping) in the two atoms involved in the hopping process. There are several effects that come with this generalization o~ the Hubbard model. First, the average hopping rate for an electron will now depend on the band tilling. as is seen b~ taking the average of Eq. I i.;bk Second, ne~, interactions are generated, as !he hopping terms in Eq. {13a) will now invoh'e four and six fermion operators. These have the potential of giving rise to ne~.~ dynamical effects, absent in the Hubba;d model. Finally, the Hamiltonian equation t 13) will. in general, not be electron hole symmetric (unless t,, -- tel.

368

J,E. Hirsch / Physica B 199&200 (1994) 366-372

To assess the importance of these effects for real systems one faces two tasks: (!) to calculate the magnitude of the parameters in the Hamiltonian equation (13) quantitatively from first principles in the various real situations of interest, and (2) to perform a many.body calculation that is able to predict reliably the properties of the system described by this Hamiltonian, and in particular the possible dynamical consequences of the new interaction terms, Concerning the second task, we have assumed in recent work that the ordering

~to > tt > t2

(14)

holds in real systems of interest, that the parameter At = t t - - t 2

(15)

is a nonnegligible fraction of an eV (0.2-0.8 eV) and that the holaping rate t2 could become very small (in particular, much smaller than At) and we have explored the consequences of these assumptions for cases where the band described by Eq. (13) is nearly full [5]. it was found that under these conditions superconductivity at high temperatures can occur, with various characteristic features that are seen in high T¢ oxides. Other characteristic features predicted by this theory are still open to experimental verification. Because the theory is applied in the regime of low carrier [hole) concentration, it is well controlled and ~e believe that the results are reliable, contrary to what would happen near the half-filled band case. Concerning the first task, we have recently performed first principles calculations [6], some of which ~,i~i be summarized in the next section. The results indicate that the assumptions listed above tend to be valid in cases where ttae effective ionic charge is small and the interatomic distances are small (i.e. densely packed, negatively charged ions).

atom we may try the Hartree wave function [7] ~bn(rt, r2) = a~(rt )a~(r2)

(17)

or the Eckart wave function [8] that allows for two different orbital exponents: ~e(rl, r2) --- a~,,(rt)a~,{r2) + a~,(rt)a,,(r2) (2(1 + $2))1/2

(18a)

S = (a,,, a,~).

(18b)

The energy is nfinimized in the first case by ~ = 11/16, yielding E = 12.86eV, and in the second case by ~ = 1.0392, ~2 =0.2832, yielding E = - 1 3 . 9 6 e V . Both wave functions improve over the Is-Is wave function given by Eq. (3), and the Eckart wave function does slightly oetter than the Is-2s wave function given by Eq. (9). The situation is rather similar for the various atomic orbitals. Figure 1 shows the orbital exponents in the Eckart wave function that minimize the energy with two electrons in the Is, 2p and 3d orbitals versus ionic charge Z. In all cases, it is seen that one of the exponents is rather close to the one appropriate to the singly occupied orbital (~ = Z, Z/2 and Z/3 for Is, 2p and 3d, respectively) while the second one is substantially smaller, implying a more diffuse orbital. As an illustration, Fig. 2 shows contours for the p orbitals corresponding to the t',,o exponents of the Eckart wave function for Z = 2 [~j = 1.06, ~2 = 0.54). It i. reasonable to expect variation in the hopping amplitude %r various occupation~ circa such a large variation in t i e effective atomic orbitals. To calculate the hopping amplitudes, we regard the hopping process as a resonance between the different ionic configurations of the neighboring atoms. For the ex~gen mn configurations relevant to high T~ oxides we have

0

0~-~0 0

0

0~0

0

0

,

[19a)

2. First principles calculations in quantum chemical calculations, it is more usual to work with Slater orbitals with variable exponents rather than with hydrogenic orbitals with fixed exponents. The Slater ls orbital

a,(r) =

e-''

(16~

(r,, = coordinate relative to atom al coincides with the [s atomic orbital for x = Z. For two electrons in a hydrogen

0

t2

~-,0

.

llgb~

0 .

(19c1

which we denote generically as

t4R

A ~-, B.

~2()~

J.E. Hirsch / Pltysim B IYYdi200 (1994) 366-372

369

O.Ot.~....I....~....~....~...;.J

1.2

1

1.6

1.4

2

1.8

I:

Fig. 2. Contours of p wave function for orbital cxponcnts (solid lines) and x2 = 0.54 (dashed lines) appropriate to the case Z = 2, for two oxygen atoms at nearest neighbor positions in a Cu--0 plane. We plot cor.;ours qwhcrethe wace function has an amplitude of0.2.0.4.0.6 and 0.8 of its maximum value. 2, = 1.06

0.0

1

1.5

2

2.5

3

z

3.5

4

We calculate the expectation value of the the symmetric and antisymmetric linear combinations. (31a)

2.0 r” s s CL 2 2 i; 2

1.5 and obtain the hopping these energies:

1.0

amplitudes

from the splitting

of

0.5

0.0

2

1

4

3

6

5

P Fig.

1 Orbital

(Eq.

I 1X)) that

(a) Is, (b) 2p

t3b>onents minimiTe

of

the

the energy

Eckrirt

svace

functacrn

for two electrons

in a

and tc) 3d orbital as a function of the ionic charge

Z (solid lines). The exponents of the Hartree wave function (Ey. (17)/.

x, m

well

as the cxponcnt\

occupied orbltrtl1.7 rc\pcctl!ely.

appropriate

to the smglq

)I) ;irc al\o st~otvn BSdashed and dotted line\.

For io and il. this i; equivalent to calculating the sp!?ttiRg @‘the lowest energy state3 of even and odd symmetry. Figure 3 shows the hopping amplitudes versus interatomic distance for electrons in Is orbitals taking for the \vave functicn of the two-electron atcams: (a) fixed IS artree wave function :>rbitals (Eq. (3)); (b! the optimal (Eq. (17)); (c) the optimal E&i. ( w.ase fur?c!im (Eq. ii 8)). it can be seen that as the atomic wave function electrons becomes more ascurate. the difkence ping amplitudes increases.

for two in hop-

370

J.E. Hirsch / Physica B 199&200 (1994) 366-372

....

I'''

"~I

....

I ....

\

I

(,. z

s

....

3.0

I ' ' ''7

t

i

a-l,

....

I ' ''~,.I

....

I ....

'n'

2,5

I ....

(a) Z-4

-

2,0

4

1,5

'~,

"%,

L'"

1.0

-

~ 1:0,~-;

0.5

1

" R .I¢£~1;t

'~:..'d:2

I

I

O0 -

I

2

"01~"~'~"~~

4

3 R (a.u.)

-

5

~ ,

0"00

....

5

I ' ' ~'~1 . . . .

•

'

"~.\ ~',\

",

-

4

I .... I .... (b) Z-1

x\

I

. . . .

2

. . . .

I

I .... 2.5

(x=l, ~-II/16

"'~,~

,

I .

.

, ~ . - - L

4 O R "a,u,)

B

3,0

6

,

. . . .

I

. . . .

I

~'

.....

8

. . . .

I

10

. . . .

(b) Z=2

to

"

2.0

0

1.5 3

1.0 2

0.5 1 0

'

I

x

I

.,. 0.0 0

1

0

8

2

I ....

''''

3 R (~.u,)

--\1 . . . .

43

/"

"\\\\t.~

_

I''''1

,

0

1

....

I ....

cxl'l,04, ~2"0.28.

"--

'

i ....

2

%.I ....

3 R [~.u,)

t

t ....

I-uY'7. ,~

4

5

6

Fig. 3. Hopping amplitudes for one, two and three electrons in ls orbitals Ito, tt and t:) versus interatomic distance. Z = !. In [ah the orbit,d exp~ments for the two-electron atom arc :ake,', ,.~ he the same :,4 for the one electron atom: in (b) and Ic), the orbital exponents appropriate to the Hartrce (Eq. 117)1 and Eckart (1-N. I18)) ~avc functions arc used Note that LM = r~ - t: increases as the wave f u n c t i o n s become nlorc

a t O L l f;|tt?.

4 6 R (~.u.)

8

I0

Fig. 4. Hopping amplitudes for electrons in p orbitals in the geometry appropriate to oxygen p~ orbitals in the planes of high T¢ oxides [the orbitals form a 45 angle with the line joining the atoms) versus interatomic distance, fu, two values of the effective ionic charge Z. The Eckart wave function was used for the two-electron case.

"x.~

_ I: 2

0 ~

....

6

(el Z=l

\

5

5

4

2

The variation of hopping amplitudes with charge occupation depends strongly on the effective nuclear charge Z. As an example, Fig. 4- shows hopping amp!itude~ versus distance for p orbitals in the g e o m e t r y a p p r o p r i a t e to high T, oxides and two values of Z. For large Z, the hopping amplitudes are all very close to each other and the ordering is opposite to that given by Eq. 114). F o r small Z instead, tlle ordering Eq. 114) holds lover an mteratomic distance range that increases as Z decreases) and the difference between h o p p i n g amplitudes is appreciable. Several other examples for both s and p orbitals are given in Ref. [6_':. The rativ At t, is found to be. approximately independent of interatomic disrmce and to increase markedly as the ionic charge decreases. ~"or lhe case of p orbitMs, this is shown in F i g 5. It was assumed it, our work on hole superconductivity [5] thai this rati~ was approximately

J.E. Hirsch / Ph.rsica B 199&200 (1994) 366 372 . . . .

I

. . . .

I

. . . .

I

. . . .

I

. . . .

6

r~

4

2

Z-1.5

-

Z-2

0

'

'

'

'

[

0

. . . .

[

2

~

i i

4 R

~ 6

8

10

(a.u.)

371

case of p orbitais and effective nuclear charge Z = 12, we find U = 4.74 eV, At = 0.225 eV, te = 0.043 eV. Is a small value of the effective nuclear charge Z reasonable for high T¢ oxides'?. Siater's lules [9] would suggest Z = 4.2 for an electron in the, negative ion O - . However, as shown by the calculations in Re('. [10], negative ions such as O - should be thought of as "having Z electrons bound as in the neutral atom with the (Z + i)st electrons occupying a far more diffuse orbital" [10]. The ( Z + 1)st electron thus would see an "effective Z" that is much smaller than the one given by Slater's rules. T o definitely answer this question, however, requires a calculation of the hopping amplitudes that takes into account correlations between all the electrons in the outer shell of the oxygen anions.

Fig. 5. Ratio of At = h - t2 to the single hole hopping amplitude Q versus interatomic distance for the p orbitals of Fig. 4 for several values of the effective ionic charge. 3.

'"'

150 --

'

\

'

I . . . .

\ \

\

I

. . . .

I

.....

U-5eV. V-O ~t-O.1875sV \

Locahzed

th-O.O3sV

-

I

J

. . . .

--15

i

.,

"

I ~

I00

50

Suporconductlnq

0.05

~t

0.1

0.15

- -

0.2

5

0.25

nh

Fig. 6. Critic:,.1 lemp,:rature versus hole conaentratioa (solid line~ for a repulsive Hubbard model where the hopping amplitudes vary with charge occupation. Parameter ~alues are given m the tigure (th = t:l. The behavior of lhe superconducting coherence length ,{ and the effcclive mass cnhancemenl ~ersus hole concentration are also shown ~dashcd and dashed dolled line, re',pccli~ely, right scalc).

independent ,ffdista,,.:: and could be substantial]} larger than unit}. Results for the case of s orbitals are sirndar. Figure 6 shows the critical temperature versus hole concentration obtained from BCS for typical parameters used in our work on hole superconductivity. The parameters m Fig. 6 are consistent with the ones obtained from the first principlc~ calculation. For example, for the

Conclusions

The results of the first principles calculations show that hopping amplitudes of electrons between neighboring ions can depend strongly on the instantaneous charge occupation of the two atoms involved in the hopping process. The Hubbard Hamiitonian equation ( i ) purports tc~ describe a conduction band with atomic configurations with zero. one and two electrons in the orbital, but assumes that \he hopping rate t i:, independent of the charge occupation of the orbital. It will thus fail to describe electronic con elations that develop due to the variation of hopping amplilude with charge occupation. The failure will be most severe in cases where the effective ionic charge is small, as these effects become quantitatively more important, and when the band is more than half-full, as double occupancy becomes unavoidable. It appears that the domin::nt new correlation effect that occurs due to this variation in the hopping amplitudes is a tendency to pairing of carriers when the Fermi level is close to the top c,¢the band Thi~ was found ;n Ref [5] and independently by other workers [-11]. The reason is that a single hole hops with. amplitude re. ~hile in the ]J'l~,.}K. Ill..r,.

tgl

~IIIkPllI~I

Ill ?t~

Ila

II'']el'lll

~

u'l"

l . . . . . . . . . . . . . . . . . . .

tort = t2 "- At. Pairingofho;es is thus favored due tothe resulting gain in kinelic energy. Rf the gain is sufficient to overcome the ordinary C o u l o m b repulsion bet~veen holes, superconductivity will result. A characteristic feature of the normal state of such systems is that the Hall coefficient will tend to bc poslti,~e, indicating the holedike nature of the charge carriers. It has been noted long ago by Chapnik [12] th;t~ ~he occurrence of superct,nductivity in elements and ,.omp,mnds ix strongly coq'dated ,~i~h a positive Hall coef~cient in the normal state. The >ituation for the elements i~

J.E. Hirsch / Physica B 199&200

372

(1994)

366-372

References Cl1 I. Hubbard, Proc. Roy. Sot. London A 276 (1963) 238.

PI l’he Hubbard Model: A Reprint Volume, ed. A. Montorsi

o o

.

ww

-0.2

0.2

l/R fi0”

A secO/dl

*

Fig. 7. Superconducting critical temperature of the elements plotted versus the invcrsc Hall cocfficicnt at low tempcraturcs and high fields. Data for the Hall coefficients were obtained from Ref. [13].

illustrated

in Fig.

7. Is it possible

that this correlation

that occurs in nature is due to the variation in hopping amplitudes Within

with

charge

of superconductivity empirical

occupation

the conventional

discussed

electron-phonon

[I41

no explanation

here?

mechanism for Chapnik’s

rule has ever been proposed.

In closing, we note that recently observed changes in the optical absorption

in the visible frequency range of

high T, superconductors

with temperature

pected for this mechanism

both in the normal [I61 and in

the superconducting

[lS]

are ex-

[I 71 state.

1%orld Scientific, Singapore, 1992). PI Even putting one electron fully in the 2r orbit,ti and the other in the Is orbital yields lower energy than both electrons in the 1s orbital: - 10.70eV for ;he singlet, - 11.89 eV for the triplet state. H. c41 Hotop and W.C. Lineberger, J. Phys. Chem. Ref. Data 4 (1975) 539 PI J. Hirsch and F. Marsiglio, Phys. Rev. B 39 (1989) 11515; Phys. Rev. B 45 (1992) 4807; F. Marsiglio and J.E. Hirsch, Phys. Rev. B41(1990)6435; Phys. Rev. B44(1991) 11960; F. Marsiglio, Phys. Rev. B 45 (1992) 956; J.E. Hirsch, Physica C I58 (1989) 326; Physica C I61 (1989) 185; 179(1991) 317; Physica C 182 (1991) 277; Physica C 194 (1992) 119. J.E. Hirsch, Chem. Phys. Lett. I71 (1990) 161; Phys. Rev. I31 B 48 (1993) 3327; Phys. Rev. B 48 (lY93) 3340; Phys. Rev. B 48 (1993) 9815. L71 D.R. Hartree, The Calculation of Atomic Structures (Wiley, New York, 1957). I31 C. Eckart, Phys. Rev. 36 (1930) 878; H. Schull and P.O. Lowdin, J. Chem. Phys. 25 (1956) 1035; J.N. Silverman, 0. Platas and F.A. Matsen, J. Chem. Phys. 32 (1960) 1402. c91 J.C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill, New York, 1960) ch. 9. cw B.H. Botch and T.H. Dunning, J. Chem. Phys. ‘I:,(1382) 6046. Cl11 R. Micnas, J. Ranninger and S. Robastkiewicz, Phys. Rev. B 39 (1989) I 1653; J. Appel, M. Grodzicki and F. Paulsen, Phys. Rev. B 47 (1993) 2812; J. Kasperzyk and H. Buttner, Z. Phys. B 87 (1992) 155; K.I. Wysokinski, Physica C 198 (1992) 87. [IXl 1.M. Chapnik, Sov. Phys. Dokl. 6 (1962) 988: Phys. Lett. A 72 (1979) 255; J. Phys. F13 (1983) 975; Phys. Stat. Sol. B I23 (1984) Kl83. [I31

New York.

Acknowledgements This Physics

work

was supported San Diego.

Dr. R.L. Martin cussions.

and Dr.

1573) and refcrcnces therein.

1141 See. for example, Superconductivity,

and the Committee

of CJifornia,

C.M. Hurd. The Hall Effect in Metals and Alloys (Plenum,

by the Department on Research,

The

author

P.R. Taylor

of

cel Dekker,

is grateful

to dis-

State Commun. [lh]

ed. R.D. Parks (Mar-

1969).

[ 151 1. Fugol. V. Samovarov, A.

University

for helpful

New York,

Ratner and V. Zhuravlev, Solid

86 [ 1993) 385.

J.E. Hirsch and F. Marsiglio,

Physica C 195 (1992) 355.

[I 71 J.E. Hirsch. Physicd C 199 (1992) 305; Physica C 201 (1992) 347.