- Email: [email protected]
Characterizing unpaired electrons from the one-particle density matrix
Chemical Physics Letters 372 (2003) 508–511 www.elsevier.com/locate/cplett
Characterizing unpaired electrons from the one-particle density matrix Martin Head-Gordon
*
Department of Chemistry, University of California, Berkeley, CA 94720-1460, USA Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720-1460, USA Received 27 January 2003; in final form 21 February 2003
Abstract A new definition of the unpaired electrons in a molecule is proposed, which derives from the one-particle reduced density matrix. It yields lower estimates of the number of radical electrons than the widely discussed Ôdistribution of effectively unpaired electronsÕ, with a maximum possible difference of a factor of two. Unlike the existing definition, the new definition cannot yield numbers of unpaired electrons higher than the total number of electrons, and also recovers the intuitively expected result for the dissociation of O2 . Ó 2003 Elsevier Science B.V. All rights reserved.
Rather like aromaticity, radical and diradical character in molecules is intuitively understood but not necessarily unambiguously defined, either experimentally or theoretically from quantum mechanics [1–3]. On the theoretical side, one particularly interesting proposal is that a density matrix of Ôeffectively unpaired electronsÕ, D, can be defined from the spinless one-particle reduced density matrix, P, as [4–11] D ¼ 2P P2 :
ð1Þ
Here we have assumed that both matrices are expressed in an orthonormal basis, although it is straightforward to generalize to a nonorthogonal basis. D can be characterized by plots, or by *
Fax: +1-510-643-1255. E-mail address: [email protected]
population analysis, just like P. The trace of D defines a total number of effectively unpaired electrons, ND ND ¼ TrðDÞ:
ð2Þ
The properties of D have been discussed in detail [10], and include: (a) The eigenfunctions of D are the natural orbitals (of P). (b) D is positive semidefinite, and for an N-electron system, the number of effectively unpaired electrons ND is bounded from above and below according to 0 6 ND 6 2N ;
ð3Þ
ND has been used to measure diradical character along a reaction coordinate for the Cope rearrangement [9], and D itself has been parti-
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00422-6
M. Head-Gordon / Chemical Physics Letters 372 (2003) 508–511
509
tioned into atomic contributions by both Mulliken and Bader schemes [9,11]. In the natural orbital basis V which diagonalizes P as P ¼ VnVy , we can express the density of effectively unpaired electrons as D ¼ VfD ðnÞVy ;
ð4Þ
where n is the (diagonal) matrix of eigenvalues, and fD is a function that acts on each individual eigenvalue n according to fD ðnÞ ¼ 2n n2 :
ð5Þ
The function fD thus converts each natural orbital occupation number n into a corresponding unpaired electron distribution number, fD ðnÞ. Summing over all M natural orbitals gives another expression for the total number of effectively unpaired electrons ND ¼
M X i¼1
fD ðni Þ ¼
M X
2ni n2i :
ð6Þ
Fig. 1. A plot of several possible mappings between natural orbital occupation numbers of the one-particle reduced density matrix and the corresponding numbers of unpaired electrons. The dark solid curve is the definition of the distribution of effective unpaired electrons, D, defined by Eq. (1). It overemphasizes radical character in the sense that a natural orbital occupation number n can yield more than n unpaired electrons. The dashed curve is the alternative definition of unpaired electrons, U, defined by Eq. (8). From this definition, an occupation number n cannot contribute more than n unpaired electrons. The light gray solid curve corresponds to Eq. (18), mentioned at the end of the Letter.
i¼1
We observe that doubly occupied (n ¼ 2) and empty orbitals (n ¼ 0) contribute nothing to D or ND while singly occupied orbitals (n ¼ 1) yield a maximal contribution, consistent with those belonging to fully unpaired electrons. The purpose of this note is to suggest an alternative measure of unpaired electrons and their distribution in a molecule, which offers some clear advantages over D. As discussed above, the mapping fD ðnÞ has well-defined integer values at n ¼ 0; n ¼ 1 and n ¼ 2, that we accept as intuitively correct. However, the non-integer values that connect these three points via a continuous function must be viewed as somewhat arbitrary. In principle a different continuous function of the natural orbital occupation numbers that also passes through those three points could equally well be used instead, if there is good reason to do so. To begin our discussion about why there is indeed good reason to consider an alternative form, we plot fD ðnÞ in Fig. 1, along with another function, that we denote as fU ðnÞ. The function fU ðnÞ is the basis of our alternative definition of the unpaired electron distribution and is given by
fU ¼ minðn; 2 nÞ ¼ 1 absð1 nÞ;
ð7Þ
fU ðnÞ represents, in our view, the simplest hypothesis for how the number of unpaired electrons changes with orbital occupation number. We just interpolate linearly between the three points for which we have well-defined values. Consequently the number of unpaired electrons increments as the occupation number itself for (effectively empty) orbitals with very small occupations. It also increments as the difference from 2 for (effectively occupied) orbitals with large occupation numbers. From this function, we obtain an unpaired electron density matrix U U ¼ VfU ðnÞVy
ð8Þ
and an associated measure of the number of unpaired electrons: nU ¼ TrðUÞ ¼
M X
fU ðni Þ
i¼1
¼
M X
minðni ; 2 ni Þ;
ð9Þ
i¼1
where M is the dimension of the one-particle basis.
510
M. Head-Gordon / Chemical Physics Letters 372 (2003) 508–511
From Fig. 1 it is evident that fD ðnÞ always equals or exceeds fU ðnÞ in magnitude. The relative difference is greatest close to (but not at) the endpoints at 0 and 2, where they differ by a factor of two. Specifically, for a very small occupation number d, the contribution to the number of unpaired electrons is 2d via fD ðnÞ, but just d via fU ðnÞ. This increment of 2d via fD ðnÞ is the origin of the upper bound of 2N for nD given in Eq. (3). It comes from distributing the electrons into many orbitals all with very small occupation numbers, because this gives contributions to nD that are maximal (double) relative to the occupation numbers themselves. This unphysical result is remedied by the alternative mapping fU ðnÞ which trivially satisfies ð10Þ
0 6 nU 6 N :
The new upper bound results from the fact that contributions to the number of unpaired electrons nU cannot exceed the occupation numbers themselves, as is clear in Eqs. (7) or (9). Thus the main reason to consider the new definition of unpaired electron distribution U is the fact that it does not exaggerate the contribution of small occupation numbers. By contrast we have shown that D systematically over-estimates the number of unpaired electrons, by a factor which can be as large as 2. The use of U is also consistent with the widely assumed measure of diradical character from the occupation number of the first natural orbital that is empty in the reference determinant (i.e., the natural LUMO). A slight formal disadvantage of U is that fU ðnÞ is not a smooth function (it is cusped at n ¼ 1), and therefore must be evaluated directly in the natural orbital basis, rather than being expressible as a polynomial in P like D (as in Eq. (1)). Let us next consider the application of this definition to the special case of unrestricted Hartree–Fock (UHF) theory (Na P Nb ). The corresponding a and b orbitals of UHF theory [12] have diagonal overlaps ki and zero off-diagonal overlaps. The diagonal overlaps connect to natural orbital occupation numbers ni ¼ 1 þ ki ;
i ¼ 1 . . . Nb ;
ð11Þ
ni ¼ 1;
i ¼ Nb þ 1; . . . ; Na ;
nN þ1i ¼ 1 ki ;
i ¼ 1; . . . Nb :
ð12Þ ð13Þ
Following [10], these occupation numbers yield Nb paired sets of doubly degenerate eigenvalues in D, with values: fD ðni Þ ¼ fD nðN þ1iÞ ¼ 1 k2i ; i ¼ 1; . . . ; Nb ð14Þ in addition to (Na Nb ) eigenvalues which are 1. Direct summation then yields: NDUHF ¼ N 2
Nb X
k2i :
ð15Þ
i¼1
Application of Eq. (7) also yields a paired structure for the eigenvalues in the new definition, U: fU ðni Þ ¼ fU nðN þ1iÞ ¼ 1 ki ; i ¼ 1; . . . ; Nb ð16Þ in addition to (Na Nb ) eigenvalues which are 1. Since 0 6 ki 6 1, it is evident that the paired eigenvalues of UUHF , (16), are necessarily each smaller than the corresponding paired eigenvalues of DUHF , (14), as expected. Direct summation then yields: NUUHF ¼ N 2
Nb X
ki :
ð17Þ
i¼1
Note that there is no direct connection between numbers of unpaired electrons according to the new definition, and hS^2 iUHF , by contrast with D [10]. As an important practical comparison of D and U, we re-examine the question of unpaired electrons in the dissociation of N2 and O2 , using the CASSCF occupation numbers reported previously [10]. At dissociation, nD ¼ 6 for N2 , which is consistent with the breaking of three bonds to form N atoms, in quartet states (i.e., 6 orbitals with occupation number 1). Strangely, however, nD ¼ 5 for the dissociation of O2 , where there are four degenerate p orbitals, each with occupation number 3/2, and two r orbitals with occupation number 1. Yet O2 at dissociation consists of two triplet O atoms, spin-coupled to make an overall
M. Head-Gordon / Chemical Physics Letters 372 (2003) 508–511
triplet, which would appear to have four unpaired electrons. Applying Eq. (9), we can see that N2 at dissociation yields nU ¼ 6 also. However, O2 at dissociation yields nU ¼ 4, which is the physically sensible result. The difference is a simple consequence of the different mappings of n ¼ 3=2 shown in Fig. 1, which again underscores the fact that the number of unpaired electrons tends to be overestimated by the definition of nD but not by the alternative, nU . The fact that these numbers of unpaired electrons are exactly integers at dissociation is a result of the limited correlating space [10]; in general we should expect a non-integer number because of remaining atomic correlations at dissociation. In conclusion, we have presented an alternative way of defining a distribution of unpaired electrons U from the one-particle density matrix that has some advantages over the existing proposal, D. U can be plotted and analyzed in the same way as D, and we recommend it for this purpose; we intend to present detailed applications to diradicaloid systems elsewhere. We observe, as pointed out by the referee, that functions like fD ðnÞ and fU ðnÞ also arise in constructing multireference perturbation theories [13]. In closing we emphasize that an infinity of possible alternative definitions also exist, and thus we cannot claim to have resolved the problem of defining unpaired electrons in molecules. For instance, another plausible, though less simple, candidate function is one that emphasizes closed shell character around n ¼ 0 and n ¼ 2, and emphasizes diradical character around n ¼ 1. Perhaps the simplest example of a function of this type is fS ðnÞ ¼ n2 ð2 nÞ2 :
ð18Þ
511
It has turning points at n ¼ 0; n ¼ 1 and n ¼ 2, as plotted in Fig. 1. Because it exaggerates diradical character around n ¼ 1, it can potentially yield a number of unpaired electrons that is greater than the number of electrons. For O2 at dissociation, it would yield 4.25 unpaired electrons with the occupation numbers used above. Acknowledgements This work was supported by the Director, Office of Basic Energy Sciences, Chemical Sciences Division of the US Department of Energy under contract DE-AC03-76SF0098. References [1] L. Salem, C. Rowland, Angew. Chem. Int. Edit. 11 (1972) 92. [2] W.T. Borden, E.R. Davidson, Acc. Chem. Res. 14 (1981) 69. [3] A. Rajca, Chem. Rev. 94 (1994) 871. [4] K. Takatsuka, T. Fueno, K. Yamaguchi, Theor. Chim. Acta 48 (1978) 175. [5] K. Takatsuka, T. Fueno, J. Chem. Phys. 69 (1978) 661. [6] R.C. Bochicchio, Theochem. J. Mol. Struct. 74 (1991) 209. [7] R.C. Bochicchio, Theochem. J. Mol. Struct. 74 (1991) 227. [8] R.C. Bochicchio, Theochem. J. Mol. Struct. 429 (1998) 229. [9] V.N. Staroverov, E.R. Davidson, J. Am. Chem. Soc. 122 (2000) 7377. [10] V.N. Staroverov, E.R. Davidson, Chem. Phys. Lett. 330 (2000) 161. [11] L. Lain, A. Torre, R.C. Bochicchio, R. Ponec, Chem. Phys. Lett. 346 (2001) 283. [12] A.T. Amos, G.G. Hall, Proc. R. Soc. London A 263 (1961) 483. [13] P.M. Kozlowski, E.R. Davidson, J. Chem. Phys. 100 (1994) 3672.
Characterizing unpaired electrons from the one-particle density matrix Martin Head-Gordon
*
Department of Chemistry, University of California, Berkeley, CA 94720-1460, USA Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720-1460, USA Received 27 January 2003; in final form 21 February 2003
Abstract A new definition of the unpaired electrons in a molecule is proposed, which derives from the one-particle reduced density matrix. It yields lower estimates of the number of radical electrons than the widely discussed Ôdistribution of effectively unpaired electronsÕ, with a maximum possible difference of a factor of two. Unlike the existing definition, the new definition cannot yield numbers of unpaired electrons higher than the total number of electrons, and also recovers the intuitively expected result for the dissociation of O2 . Ó 2003 Elsevier Science B.V. All rights reserved.
Rather like aromaticity, radical and diradical character in molecules is intuitively understood but not necessarily unambiguously defined, either experimentally or theoretically from quantum mechanics [1–3]. On the theoretical side, one particularly interesting proposal is that a density matrix of Ôeffectively unpaired electronsÕ, D, can be defined from the spinless one-particle reduced density matrix, P, as [4–11] D ¼ 2P P2 :
ð1Þ
Here we have assumed that both matrices are expressed in an orthonormal basis, although it is straightforward to generalize to a nonorthogonal basis. D can be characterized by plots, or by *
Fax: +1-510-643-1255. E-mail address: [email protected]
population analysis, just like P. The trace of D defines a total number of effectively unpaired electrons, ND ND ¼ TrðDÞ:
ð2Þ
The properties of D have been discussed in detail [10], and include: (a) The eigenfunctions of D are the natural orbitals (of P). (b) D is positive semidefinite, and for an N-electron system, the number of effectively unpaired electrons ND is bounded from above and below according to 0 6 ND 6 2N ;
ð3Þ
ND has been used to measure diradical character along a reaction coordinate for the Cope rearrangement [9], and D itself has been parti-
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00422-6
M. Head-Gordon / Chemical Physics Letters 372 (2003) 508–511
509
tioned into atomic contributions by both Mulliken and Bader schemes [9,11]. In the natural orbital basis V which diagonalizes P as P ¼ VnVy , we can express the density of effectively unpaired electrons as D ¼ VfD ðnÞVy ;
ð4Þ
where n is the (diagonal) matrix of eigenvalues, and fD is a function that acts on each individual eigenvalue n according to fD ðnÞ ¼ 2n n2 :
ð5Þ
The function fD thus converts each natural orbital occupation number n into a corresponding unpaired electron distribution number, fD ðnÞ. Summing over all M natural orbitals gives another expression for the total number of effectively unpaired electrons ND ¼
M X i¼1
fD ðni Þ ¼
M X
2ni n2i :
ð6Þ
Fig. 1. A plot of several possible mappings between natural orbital occupation numbers of the one-particle reduced density matrix and the corresponding numbers of unpaired electrons. The dark solid curve is the definition of the distribution of effective unpaired electrons, D, defined by Eq. (1). It overemphasizes radical character in the sense that a natural orbital occupation number n can yield more than n unpaired electrons. The dashed curve is the alternative definition of unpaired electrons, U, defined by Eq. (8). From this definition, an occupation number n cannot contribute more than n unpaired electrons. The light gray solid curve corresponds to Eq. (18), mentioned at the end of the Letter.
i¼1
We observe that doubly occupied (n ¼ 2) and empty orbitals (n ¼ 0) contribute nothing to D or ND while singly occupied orbitals (n ¼ 1) yield a maximal contribution, consistent with those belonging to fully unpaired electrons. The purpose of this note is to suggest an alternative measure of unpaired electrons and their distribution in a molecule, which offers some clear advantages over D. As discussed above, the mapping fD ðnÞ has well-defined integer values at n ¼ 0; n ¼ 1 and n ¼ 2, that we accept as intuitively correct. However, the non-integer values that connect these three points via a continuous function must be viewed as somewhat arbitrary. In principle a different continuous function of the natural orbital occupation numbers that also passes through those three points could equally well be used instead, if there is good reason to do so. To begin our discussion about why there is indeed good reason to consider an alternative form, we plot fD ðnÞ in Fig. 1, along with another function, that we denote as fU ðnÞ. The function fU ðnÞ is the basis of our alternative definition of the unpaired electron distribution and is given by
fU ¼ minðn; 2 nÞ ¼ 1 absð1 nÞ;
ð7Þ
fU ðnÞ represents, in our view, the simplest hypothesis for how the number of unpaired electrons changes with orbital occupation number. We just interpolate linearly between the three points for which we have well-defined values. Consequently the number of unpaired electrons increments as the occupation number itself for (effectively empty) orbitals with very small occupations. It also increments as the difference from 2 for (effectively occupied) orbitals with large occupation numbers. From this function, we obtain an unpaired electron density matrix U U ¼ VfU ðnÞVy
ð8Þ
and an associated measure of the number of unpaired electrons: nU ¼ TrðUÞ ¼
M X
fU ðni Þ
i¼1
¼
M X
minðni ; 2 ni Þ;
ð9Þ
i¼1
where M is the dimension of the one-particle basis.
510
M. Head-Gordon / Chemical Physics Letters 372 (2003) 508–511
From Fig. 1 it is evident that fD ðnÞ always equals or exceeds fU ðnÞ in magnitude. The relative difference is greatest close to (but not at) the endpoints at 0 and 2, where they differ by a factor of two. Specifically, for a very small occupation number d, the contribution to the number of unpaired electrons is 2d via fD ðnÞ, but just d via fU ðnÞ. This increment of 2d via fD ðnÞ is the origin of the upper bound of 2N for nD given in Eq. (3). It comes from distributing the electrons into many orbitals all with very small occupation numbers, because this gives contributions to nD that are maximal (double) relative to the occupation numbers themselves. This unphysical result is remedied by the alternative mapping fU ðnÞ which trivially satisfies ð10Þ
0 6 nU 6 N :
The new upper bound results from the fact that contributions to the number of unpaired electrons nU cannot exceed the occupation numbers themselves, as is clear in Eqs. (7) or (9). Thus the main reason to consider the new definition of unpaired electron distribution U is the fact that it does not exaggerate the contribution of small occupation numbers. By contrast we have shown that D systematically over-estimates the number of unpaired electrons, by a factor which can be as large as 2. The use of U is also consistent with the widely assumed measure of diradical character from the occupation number of the first natural orbital that is empty in the reference determinant (i.e., the natural LUMO). A slight formal disadvantage of U is that fU ðnÞ is not a smooth function (it is cusped at n ¼ 1), and therefore must be evaluated directly in the natural orbital basis, rather than being expressible as a polynomial in P like D (as in Eq. (1)). Let us next consider the application of this definition to the special case of unrestricted Hartree–Fock (UHF) theory (Na P Nb ). The corresponding a and b orbitals of UHF theory [12] have diagonal overlaps ki and zero off-diagonal overlaps. The diagonal overlaps connect to natural orbital occupation numbers ni ¼ 1 þ ki ;
i ¼ 1 . . . Nb ;
ð11Þ
ni ¼ 1;
i ¼ Nb þ 1; . . . ; Na ;
nN þ1i ¼ 1 ki ;
i ¼ 1; . . . Nb :
ð12Þ ð13Þ
Following [10], these occupation numbers yield Nb paired sets of doubly degenerate eigenvalues in D, with values: fD ðni Þ ¼ fD nðN þ1iÞ ¼ 1 k2i ; i ¼ 1; . . . ; Nb ð14Þ in addition to (Na Nb ) eigenvalues which are 1. Direct summation then yields: NDUHF ¼ N 2
Nb X
k2i :
ð15Þ
i¼1
Application of Eq. (7) also yields a paired structure for the eigenvalues in the new definition, U: fU ðni Þ ¼ fU nðN þ1iÞ ¼ 1 ki ; i ¼ 1; . . . ; Nb ð16Þ in addition to (Na Nb ) eigenvalues which are 1. Since 0 6 ki 6 1, it is evident that the paired eigenvalues of UUHF , (16), are necessarily each smaller than the corresponding paired eigenvalues of DUHF , (14), as expected. Direct summation then yields: NUUHF ¼ N 2
Nb X
ki :
ð17Þ
i¼1
Note that there is no direct connection between numbers of unpaired electrons according to the new definition, and hS^2 iUHF , by contrast with D [10]. As an important practical comparison of D and U, we re-examine the question of unpaired electrons in the dissociation of N2 and O2 , using the CASSCF occupation numbers reported previously [10]. At dissociation, nD ¼ 6 for N2 , which is consistent with the breaking of three bonds to form N atoms, in quartet states (i.e., 6 orbitals with occupation number 1). Strangely, however, nD ¼ 5 for the dissociation of O2 , where there are four degenerate p orbitals, each with occupation number 3/2, and two r orbitals with occupation number 1. Yet O2 at dissociation consists of two triplet O atoms, spin-coupled to make an overall
M. Head-Gordon / Chemical Physics Letters 372 (2003) 508–511
triplet, which would appear to have four unpaired electrons. Applying Eq. (9), we can see that N2 at dissociation yields nU ¼ 6 also. However, O2 at dissociation yields nU ¼ 4, which is the physically sensible result. The difference is a simple consequence of the different mappings of n ¼ 3=2 shown in Fig. 1, which again underscores the fact that the number of unpaired electrons tends to be overestimated by the definition of nD but not by the alternative, nU . The fact that these numbers of unpaired electrons are exactly integers at dissociation is a result of the limited correlating space [10]; in general we should expect a non-integer number because of remaining atomic correlations at dissociation. In conclusion, we have presented an alternative way of defining a distribution of unpaired electrons U from the one-particle density matrix that has some advantages over the existing proposal, D. U can be plotted and analyzed in the same way as D, and we recommend it for this purpose; we intend to present detailed applications to diradicaloid systems elsewhere. We observe, as pointed out by the referee, that functions like fD ðnÞ and fU ðnÞ also arise in constructing multireference perturbation theories [13]. In closing we emphasize that an infinity of possible alternative definitions also exist, and thus we cannot claim to have resolved the problem of defining unpaired electrons in molecules. For instance, another plausible, though less simple, candidate function is one that emphasizes closed shell character around n ¼ 0 and n ¼ 2, and emphasizes diradical character around n ¼ 1. Perhaps the simplest example of a function of this type is fS ðnÞ ¼ n2 ð2 nÞ2 :
ð18Þ
511
It has turning points at n ¼ 0; n ¼ 1 and n ¼ 2, as plotted in Fig. 1. Because it exaggerates diradical character around n ¼ 1, it can potentially yield a number of unpaired electrons that is greater than the number of electrons. For O2 at dissociation, it would yield 4.25 unpaired electrons with the occupation numbers used above. Acknowledgements This work was supported by the Director, Office of Basic Energy Sciences, Chemical Sciences Division of the US Department of Energy under contract DE-AC03-76SF0098. References [1] L. Salem, C. Rowland, Angew. Chem. Int. Edit. 11 (1972) 92. [2] W.T. Borden, E.R. Davidson, Acc. Chem. Res. 14 (1981) 69. [3] A. Rajca, Chem. Rev. 94 (1994) 871. [4] K. Takatsuka, T. Fueno, K. Yamaguchi, Theor. Chim. Acta 48 (1978) 175. [5] K. Takatsuka, T. Fueno, J. Chem. Phys. 69 (1978) 661. [6] R.C. Bochicchio, Theochem. J. Mol. Struct. 74 (1991) 209. [7] R.C. Bochicchio, Theochem. J. Mol. Struct. 74 (1991) 227. [8] R.C. Bochicchio, Theochem. J. Mol. Struct. 429 (1998) 229. [9] V.N. Staroverov, E.R. Davidson, J. Am. Chem. Soc. 122 (2000) 7377. [10] V.N. Staroverov, E.R. Davidson, Chem. Phys. Lett. 330 (2000) 161. [11] L. Lain, A. Torre, R.C. Bochicchio, R. Ponec, Chem. Phys. Lett. 346 (2001) 283. [12] A.T. Amos, G.G. Hall, Proc. R. Soc. London A 263 (1961) 483. [13] P.M. Kozlowski, E.R. Davidson, J. Chem. Phys. 100 (1994) 3672.