J . Phys. Chem. 1988, 92, 6410-6414
6470
Chemical Reactivity in Spin-Polarized Density Functional Theory Marcel0 Galvln, Albert0 Vela, and JosC L. Giizquez* Departamento de Qdmica, Divisidn de Ciencias Brisicas e Ingenieria, Universidad Autdnoma Metropolitana-Iztapalapa,A.P. 55-534, MZxico, D. F., 09340 Mexico (Received: March 24, 1988)
Spin-polarized density functional theory is used to analyze chemical reactivity from a more general point of view, which distinguishes between the changes produced by charge transfer between the interacting species (changes in the total number of electrons,N = Nt + Ni where t refers to spin-up or a and 1 to spin-down or p) and the changes produced by the redistribution of the electronic density of each of the interacting species (changes in the spin number, Ns = Nt - Ni). It is found that the response of the system to changes in Nand the external potential is given in terms of the chemical potential, the hardness, the electronic density, and the Fukui function, while the response of the system to changes in Ns and an external magnetic field is given in terms of a new set of parameters which we have named the spin potential, the spin hardness, the spin density, and the spin Fukui function. Making use of the Kohn-Sham approach to density functional theory, it is shown that the generalized Fukui functions can be reduced to a set of spin-polarized classical frontier orbitals by imposing frozen core approximations.
I. Introduction The thermodynamic nature of density functional theory14 has provided a formal theoretical framework to several concepts such as electronegativity,1 h a r d n e s ~and , ~ frontier orbitals,b8 which are frequently used to understand chemical reactivity. In density functional theory one may write, formally, the total energy in terms of the charge density p(7) and the external potential u(7) in the form9
J%WI
= F[pl
+
S u ( 7 ) ~ ( 7 d7 )
(1)
and
(7)
=
one can rewrite eq 5 in the form dE = p dN
+ s p ( F ) &I(?)
d7
(8)
Similarly, if one considers the chemical potential as a function of the number of electrons and a functional of the external potential, one has that6
with F b l = T[PI +
veebl
(2)
where T[p]is the electronic kinetic energy functional and V,[p] is the electron-electron repulsion energy functional. The minimization of the total energy with respect to the charge density, subject to the condition that the number of electrons is fixed, N = s p ( 7 ) d7
(3)
leads to an Euler-Lagrange equation of the form
and the definition of hardnessS 9 =
(4) The solution of eq 4 leads to the ground-state charge density, from which one can determine the ground-state energy. Now, if one considers the energy as a function of the number of electrons, and a functional of the external potential, then one has that*
Since
Using a Maxwell relation for eq 8, namely3
(s),
= P(7)
(1) Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J . Chem. Phys. 1978, 68, 3801. (2) Parr, R. G.; Bartolotti, L. J. J . Phys. Chem. 1983, 87, 2810. (3) Nalewajski, R. F.; Parr, R. G. J . Chem. Phys. 1982, 77, 399. (4) Berkowitz, M.; Ghosh, S. K.; Parr, R. G. J . Am. Chem. Soc. 1985,107, 6811. ( 5 ) Parr, R. G.; Pearson, R. G. J . Am. Chem. Soc. 1983, 105, 7512. (6) Parr, R. G.; Yang, W. J . Am. Chem. SOC.1984, 106, 4049. (7) Fukui, K. Theory of Orientation and Stereoselection; Springer-Verlag: West Berlin, 1973. (8) Yang, W.; Parr, R. G.; Pucci, R. J . Chem. Phys. 1984, 81, 2862. (9) Hohenberg, P.; Kohn, W. Phys. ReD. [Sect.] B 1964, 136, 864.
0022-3654/88/2092-6470$01.50/0
(
$)u
=
(5)
one can rewrite eq 9 in the form dp = 7 d N
+ s f ( 7 ) b(?)d7
wheref(3 is called the Fukui or frontier function, because it reduces to the frontier orbitals by using the frozen core approxima t ion .6,8 The relations for the changes in the energy and the chemical potential, eq 8 and 12, are fundamental for the description of a time-independent chemical event. From these relations one can see that the electronegativity (chemical potential) and the hardness can be used as global reactivity criteria, since they are constant over all the space occupied by the molecule, while the charge density and the Fukui function can be used as local reactivity criteria, because their values depend on F, thus differentiating one part of a molecule from other regions. The relations presented up to this point correspond to the spin-nonpolarized case. The object of the present work is to derive the expressions corresponding to the spin-polarized case, in order to derive additional reactivity criteria. 11. Spin-Polarized Approach
It is well-known that one can improve the description of the electronic structure of atoms, molecules, and solids by breaking the charge density into spin components,I0 that is P(r’)
= PtV)
0 1988 American Chemical Society
+P
l m
(13)
The Journal of Physical Chemistry, Vol. 92, No. 22, 1988 6471
Spin-Polarized Density Functional Theory where f refers to spin-up or cy and to spin-down or (3. Within density functional theory, such an approach implies that the total energy will be a functional of the independent functions pt(F) and p i ( 7 ) , instead of just p ( 7 ) . An alternative set of independent functions is given by the total charge density p ( 9 , eq 13, and the spin density ps(7), which can be expressed in terms of the spin-up and spin-down densities as Ps(7) = P d r ' ) - /Jim
should be proved rigorously. For the moment we assume that the simultaneous solution of eq 19 and 20 leads to charge and spin densities (which correspond to prescribed values of N and Ns), from which one can determine the energy by using eq 18. Now let us consider the differential of the energy in terms of the functional given by eq 18, that is
(14)
Since l p + ( F ) d7
= Nt, i p i ( r ' ) d7 = Ni
(15)
one has that
From eq 18 one has that
and Thus, using eq 19,20, and 22, one can rewrite eq 21 in the form dE = where Ns will be denoted as the spin number. In this work we will make use of p ( 9 , ps(7), N, and Ns, because this way one can immediately identify the additional contributions that arise from the spin polarization. In terms of p and ps, the total energy for the system in the presence of an external potential u(7) and an external magnetic field B(7) in the z direction is given by",'* E[p,ps,u,Bl = F[p,psl
+ Ju(3
d7- P B J B ( ~ )PSV)d7 (18)
/,LN
d N + ps dNs + Jp(7)
SB(7) dr' (23)
where we have made use of the fact that pN and ps are constants. On the other hand, if one considers the energy as a function of the total number of electrons and the spin number, and a functional of the external potential and the magnetic field, one finds that dE =
(")
dN
where p B is the electron Bohr magneton. Since the total charge density and the spin density are independent functions, one has to carry out the minimization procedure taking into account the variation of the energy with respect to both. Thus, by imposing the conditions given by eq 16 and 17, one finds, in the case of the variation of p(?), that
6~(r') dr' - pBJps(7)
dN + Ns,u,B
(E)
dNs
+lp(7)
6u(r') d7-
aNS . N.0.B .
pBIpS(F)
dr' (24)
where we have made use of first-order perturbation theory to substitute (6E/6u(7))NpsB by p(7) and (6E/6B(7))Nflu by - p ~ p s ( 7 ) . The comparison of eq 23 and 24 leads to
and in the case of the variation of ps(7), that
It is important to note that we have introduced two Lagrange multipliers, pN and ps, because N a n d Ns are kept fixed during the variation. This statement is the equivalent of fixing the number of spin-up and spin-down electrons. An alternative approach would be to fix only the total number of electrons and to allow for changes in the values of the spin number Ns. In such case, eq 19 would remain unchanged, while ps in eq 20 should be set equal to zero. The procedure in which only the total number of electrons is fixed during the variation implies that the spin polarization of the ground state is determined by the final solution. Thus, the approach in which the spin polarization is fixed during the variation (by setting the value of Ns) will not necessarily lead to the true ground state of the system, unless the prescribed value of Ns corresponds to the correct multiplicity of the ground state. A symmetry-constrained variational procedure in density functional theory points toward the possible extension of the Hohenberg-Kohn approach to excited states;I3-l5 however, it
The first of these relations is the equivalent of the chemical potential in the spin-restricted case except for the fact that the derivative is carried out at a fixed value of Ns. The second relation, eq 26, can be called the "spin potential", since it provides a measure of the tendency of a system to change the spin polarization. From the point of view of chemical reactivity ps may provide very interesting information since for a fixed value of N , the different values of Ns can be interpreted as different valence states of the system. Thus, the values of the slope of the energy as a function of Ns may be used to explain some of the changes involved in the process of the interaction between different species. From eq 25 and 26 one may look at the chemical and spin potentials as functions of the number of electrons, the spin number, the external potential, and the magnetic field; then
and dps =
(10) Slater, J. C . Quantum Theory of Molecules and Solids; McGrawHill: New York, 1974; Vol. 4 . (11) von Barth, U.; Hedin, L. J . Phys. C 1972, 5, 1629. (12) Vosko, S. H.; Perdew, J. P. Can. J . Phys. 1975, 53, 1385. (1 3) Zaremba, E. In Density Matrices and Density Functionah; Erdahl, R., Smith, Jr., V. H., Eds.; D. Reidel: Dordrecht, Holland, 1987; pp 339-357. (14) Kohn, W. Phys. Reu. A 1986, 34, 731. (15) Theophilou, A. K. J . Phys. C 1978, 12, 5419.
~ S d NN
+ qss dNs + S f N S ( 7 )
6V(r') d 7 r*BJfSS(r')
where using Maxwell relations for eq 23, one has that
d7 (28)
6412
The Journal of Physical Chemistry, Vol. 92, No. 22, 198 8
and
fss(7)=
-q”-) pB
6B(7)
= N,Ns,u
(-)
(35)
aPs(7) aNS
N,u,B
The relation given by eq 29 is the equivalent of the hardness in the spin-restricted case, except for the fact that the derivative is carried out at a fixed value of Ns. Equations 30 and 31 provide additional information; qNs measures variations in the chemical potential with respect to changes in the spin number, or alternatively it measures variations in the spin potential with respect to changes in the total number of electrons. On the other hand, qss can be called the “spin hardness”, since it is similar to the hardness in the sense that it corresponds to a second derivative of the energy, however in this case with respect to the spin number. The quantities ps, qNs, and qSs can be used together with pN and qNNas global reactivity criteria since they characterize the species as a whole. The derivatives qNs and qss for a fixed value of N provide information on the behavior of the system for different valence states, just as in the case of ps. The quantitiesfSN(7),fNS(7), and&(?) are the equivalent of the Fukui function,fNN(7) for the spin-restricted case. In fact, fsN(7), in terms of the spin-up and spin-down variables, has already been used by Yang and P a d 6 to explain some aspects of reactivity in chemisorption and catalysis. Within the present approach one can see thatfNS(7) andfss(7) will also provide information about the initial response of a system to a given type of attack, since they measure the change of the total charge density or the spin density with respect to changes in the total number of electrons or the spin number. The setf”(7),fsN(7),fNs(7), andfss(7) can be used as local reactivity criteria, since their values vary from one point to another within the molecule. It is important to note that eq 27 and 28 provide very interesting information. In order to analyze these relations in terms of chemical reactivity, one should analyze first the meaning of the two external potentials. The change in the electrostatic potential, 6u(7), can be interpreted in terms of the potential generated by the species that surround the reacting molecule, giving rise to an electrostatic interaction. Similarly, the change in the magnetic field, 6B(7),can also be interpreted in terms of the magnetic field generated by the species that surround the reacting molecule, giving rise to a magnetic interaction. In such case, one should consider other directions of the magnetic field to describe the interaction energy, but even in this situation the latter would be given in terms of the magnetic moment density pB p s ( 7 ) of the reacting molecule. Thus, the response of the system to a given reactant may be analyzed in general through eq 27 and 28, by considering that the changes in the external potential and magnetic field arise from the presence of the surrounding species. Now, since the chemical and the spin potentials must be constants over all the space occupied by the molecule, one can see that when different atomic species interact to form a molecule. (16) Yang,W.; Parr, R. G . Proc. Natl. Acad. Sci. U.S.A. 1985,82. 6723
Galvfin et al. there are two equalization processes: the chemical potential equalization, which is achieved by charge transfer, and the spin potential equalization, which is achieved by “spin transfer”. The latter corresponds in reality to rearrangements of the spin-up and spin-down electronic charge distributions that lead to changes in the values of NS for the atoms in the molecule. Thus, independently from the important contributions to the bond formation that arise from the external potential and magnetic fields, one can expect in terms of the global quantities pN and ps that there may be cases in which the driving force comes from large differences between the chemical potentials of the reacting species, there may be cases in which the driving force comes from large differences between the spin potentials of the reacting species, and there may be cases in which both are important. Now, in order to see the importance of the local response functions in eq 27 and 28, one may adopt the point of view of the postulate of Parr and Yang,6 which establishes that chemical reactions are preferred from the direction which produces maximum initial chemical potential response of a reactant. One can see that not only the local values ofy”(7) can be used to define the reactive site, but there may be cases with an initial large value of ldpNl as a consequence of a large local value offSN, instead Off”. This observation is in agreement with facts, sincefSN is related to changes in the spin density with respect to the total number of electrons, and it is well-known that chemical species like the free radicals, which present large local values of ps(7) because of the unpaired electron, are at the same time highly reactive. On the other hand, there may also be reactions governed by large initial values of JdksI. In this case the reactive sites would be localized at the points where fNs(7) or fss(r’) is large. Thus, we have seen that by extending the energy density functional to the spin-polarized case and by imposing that the variation of the energy be carried out at fixed values of N and Ns, one can separate the reactivity criteria associated with changes in the number of electrons into reactivity criteria that measure on one hand changes with respect to the total number of electrons for a fixed value of Ns and on the other hand changes with respect to the spin number for a fixed value of N . Such an approach introduces new global and local quantities which can be associated with the flexibility of a given molecule to change its valence state in order to react with another molecule. 111. Kohn-Sham Approach and the Frozen Core Approximation
In order to have a better understanding of the quantities derived in the previous section, it is convenient to make use of the Kohn-Sham approach to density functional theory. An immediate advantage of this formalism is that one may carry out the differentiation explicitly in terms of the occupation numbers. Now, the Kohn-Sham functional for a system, in the absence of an external magnetic field, in the spin-polarized case can be written in the form”
where
T s [ P ~ , P=I I ~Cn,,~9,,*(9[-i/2V219,u(~) d7 (37) L
T
I
is the noninteracting kinetic energy functional, c = occupation of the spin orbital Q,, P O V )
t, 1, n,, is the
= Cn,uQ,,*(7) 41m
(38)
l
and the external potential contains only the contribution from the atomic nuclei of the system. Note that we have made use of the functions Pt, p , instead of p , ps because it is more convenient at the moment. Now, the minimization of the total energy with respect to the spin orbitals, subject to the orthonormality constraints, leads to (17) Kohn. W.; Sham, L. J. Phys. Reo. 1965. 140. A I 133
The Journal of Physical Chemistry, Vol. 92, N o . 22, 1988 6473
Spin-Polarized Density Functional Theory one-electron equations of the form
P:
k!hV2 + u(r3 + uZdr314iu(r3 =
ddr3
(39)
(u:~(?)= (6V',[pf,p1]/6pu(r3),), which have to be solved selfconsistently to determine eiu and &,(?). It should be noted that since the occupation numbers are kept fixed during the variation, one can always select their values to give the correct values of N and Ns, since N = C(nit + nil)
(40)
NS = C(nit - nil)
(41)
i
and i
In order to calculate lN and ps, one can make use of the results of Perdew, Parr, Levy, and Balduz,18 who have shown that the highest occupied orbital energy of the Kohn-Sham formalism is equal to the chemical potential. However, in the spin-polarized case presented here, one has to take into account that the derivatives with respect to N are performed with Ns fixed and the derivatives with respect to Ns are performed with N fixed. Thus, in the first case, for the ground state of a system with (N 6), Ns such that 6 > 0, one would have to place 6/2 electrons in &axt+l and 6/2 electrons in $maxj+l to keep NS constant, so that
=
x[
'LUMOf
= '/z(wLSf + Pi)
- 'LUMOi 2
+ 'HOMOt
- tHOMOl 2
]
(51)
Equations 46 and 5 1 provide very interesting information, because while the chemical potential depends on the averages of the orbital energies corresponding to the spin-up and spin-down LUMO and HOMO, the spin potential depends on the average differences between the orbital energies of the spin-up and spin-down LUMO and HOMO. Thus, from the point of view of chemical reactivity, one can see that not only the relative values of the orbital energies of the HOMO and LUMO of the reacting species are important but also the internal splitting of the spin-up and spin-down orbital energies of the H O M O and LUMO of each reacting species. Now we focus on the total charge density and spin density derivatives. First we note that following the same arguments of Yang, Parr, and Pucci,8 one can prove the following functional relationships: (a) For a system with N 6, Ns = constant
+
+
PT
=
tmaxt+l
and
w i = tmaxi+l
(42)
Recalling that in general pt = (dE/dNt), and pi = (dE/dNi), and making some algebraic manipulations in connection with the change from the set of independent variables Nt, N J to the set of independent variables N , N s , one finds that for 6 > 0 p& = 1/2(€LUMOt
+ 'LUMOl)
= constant, Ns
(43)
+
where LUMO = max+l is the lowest unoccupied molecular orbital. Similarly, for 6 < 0, one would have to remove 6/2 electrons from and 6/2 electrons from to keep NS constant; then
= and therefore, for 6
and w i = €maxi
(44)
MCHOMO~ + HOMO^)
(45)
tmaxt
0
+
L
HOMO?
. I
In order to calculate l sone , should take into account that the total number of electrons must be kept constant. Thus, for a system with N , (Ns hS) such that hS > 0 one has to remove 6s/2 electrons from &axl and place them in &,axt+l so that
+
= ELUMOt and and therefore, for 6s
pi
=
'HOMO4
PLSf = 1/2(€LUMOt - 6HOMOl)
(48)
Similarly, for hS < 0, one would have to remove dS/2 electrons from and place them in thus PT
=
€HOMO?
and therefore, for SS
and P i =
eLUMOl
PI(?)
=
i= 1
16$(i?12+ (6/2)l&UMOt(?))12
(58)
l6J(r3l2+
(59)
HOMO1
c
i= 1
(6/2)1d'hMOl(7))12
Making use of the frozen core approximation, which in the present context implies that the terms corresponding to the derivatives of the Kohn-Sham orbitals with respect to the number of spin-up or spin-down electrons are neglected, one finds that
(49)
0 governs nucleophilic attack, while the case 6 = 0 governs neutral (radical) attack. The generalization to the spin-polarized case establishes that not only large values of the spin-up and spin-down HOMO and LUMO may define the reactive site but also large differences between the spin-up and spin-down HOMO and LUMO may be responsible of chemical reactivity at a given point within the molecule. Following the same procedure that led to eq 62-67, one can derive the expressions forfNs(7) andfss(?) for the case N = constant, Ns + 1 3 ~ .Thus, one finds that for 6,
for 6,
>0
