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Comment on ‘Characterizing unpaired electrons from one-particle density matrix’ [M. Head-Gordon, Chem. Phys. Lett. 372 (2003) 508–511]
Chemical Physics Letters 380 (2003) 486–487 www.elsevier.com/locate/cplett
Comment on ÔCharacterizing unpaired electrons from one-particle density matrixÕ [M. Head-Gordon, Chem. Phys. Lett. 372 (2003) 508–511] R.C. Bochicchio
a,* ,
A. Torre b, L. Lain
b
a
b
Departamento de Fısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pab. 1 – Ciudad Universitaria, Buenos Aires 1428, Argentina Departamento de Quımica Fısica, Facultad de Ciencias, Universidad del Paıs Vasco, Apdo. 644, Bilbao E-48080, Spain Received 9 June 2003; in final form 8 September 2003 Published online: 7 October 2003
Abstract A recent theoretical study published in this journal has proposed a new definition of the unpaired electron density in molecular systems. This comment attempts to clarify the difference between the concept reported there and that of ÔeffectivelyÕ unpaired electron density. Ó 2003 Elsevier B.V. All rights reserved.
Since the early introduction by Takatsuka et al. [1] and Takatsuka and Fueno [2] the effectively unpaired electron density has been subject of several studies [3–6]. However, there are some misunderstandings about this matrix, may be because of its name ‘effectively’ unpaired electron, as could be inferred from [7]. This report concludes that a contradiction would arise because in some cases the trace of the effectively unpaired electron density matrix Nu exceeds the total number of electrons N . If this interpretation were right, the effectively unpaired electron matrix u should not be as useful as was considered in the near past. In this Comment
we attempt to clarify the physical meaning of this matrix and the boundary values of its trace. For such a goal let us introduce the natural definition for u matrix that expresses the lack of idempotency of the spin-free 1-particle density matrix 1 D (1-RDM) [1–6] as X 1 i1 k uij ¼ 21 Dij Dk Dj ð1Þ k
and so trðuÞ ¼ 2N trð1 D2 Þ because trð1 DÞ ¼ N . In a more explicit manner, it can also be recognized as part of the master equation (particle-hole conservation equation), Þ; 2N ¼ trð1 D1 DÞ þ trð1 D1 D
*
Corresponding author. Fax: 54-11-45763357. E-mail address: [email protected] (R.C. Bochicchio).
ð2Þ
where the first and second terms of the RHS of Eq. (2) are the statistical averages of the number of
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.09.035
R.C. Bochicchio et al. / Chemical Physics Letters 380 (2003) 486–487
particles [8] and holes in the system [3], respectively; the last one is related to the u matrix by Þ: Nu trðuÞ ¼ trð1 D1 D
ð3Þ
Eq. (2) is a consequence of the hole-particle nonseparability in many body systems at correlated level showing that the boundary values for Nu are, 0 6 Nu 6 2N . In order to proceed to clarify the physical meaning of the u matrix and its boundary values we will make use of the structure of this matrix which can be expressed by 2
u ¼ ð1 Dspin Þ D; 1
spin
ð4Þ
where D is the 1-particle spin density matrix or simply the spin density matrix, expressed as the difference 1 Dspin ¼ 1 Da 1 Db , where 1 Dr ; r ¼ a; b stands for the spin up (a) and spin down ðbÞ one particle density matrices, respectively [6]. D ¼ Da þ Db collects the total deviation from idempotency as contributions of the spin up and the spin down clouds, respectively, defined by ð1 Dr Þ2 ¼ 1 r D þ 12Dr [6]. It can be noted that the natural matrix for describing the net unpaired particles density is the spin density matrix as has been well established [9] and not the u matrix, i.e., trð1 Dspin Þ ¼ trð1 Da Þ trð1 Db Þ ¼ N a N b where N a and N b are the number of a–electrons and b–electrons in the system, respectively. Thus, there is not any contradiction between the physical meaning of u matrix and its boundary values. However, in order to complete the analysis, let us inspect the values for Nu as a consequence of such structure. The upper value Nu ¼ 2N stands for a configuration in which all orbital occupation numbers are fni 1; i ¼ 1; . . . ; Kg where K is the size of the basis set, which indicates a highly ‘‘diluted’’ state or having a great amount of holes in the configuration, supporting an unphysical situation; the lower bound, Nu ¼ 0, is the most ‘‘condensed’’ electron configuration, i.e., the first N2 orbitals are doubly occupied while the remaining ones are vacant. These results seem to indicate that the upper bound describes a very disordered configuration while the lower one describes an ordered one. Another important value for Nu is when Nu N . A simple calculation from Eq. (4) shows that this condition is fulfilled by states having the highest spin projection, i.e., Sz ¼ N2 . This case at Hartree–Fock (Restricted
487
Open Shell HF-ROHF) level of description results in u ¼ 1 Da 1 D thus Nu ¼ N . At correlated level Nu ¼ trð1 Da Þ 12trðDa Þ ¼ N 12trðDa Þ which is also an improbable physical situation except in a simple system like hydrogen molecule. These results show that Nu increases as long as correlation and even delocalization are present in the system [10]. It can be noted that values Nu > N are not consistent for molecular systems even in nonequilibrium points defining a chemical rearrangement on the potential hypersurface because atoms preserve their fundamental electronic configuration during a chemical reaction and thus ‘‘extreme’’ states are not involved in such transformations. Therefore, u matrix so defined is a measure of the ‘‘deformation’’ of the ordered population when passing from Hartree– Fock reference states to correlated states in which orbitals are not double occupied. Because such occupations are not observable quantities the physical meaning of this matrix is related to valence concepts through the hole populations as demonstrated in [6]. Singlet states at correlated level show Nu 0 but not null which indicates an nonuniform orbital occupation due to correlation effects. Thus u cannot be considered as an net unpaired electron density and it can be interpreted as local space unpaired density (cf. discussion in [3]) or, as widely used in the literature, ‘‘effectively’’ unpaired electron density matrix.
References [1] K. Takatsuka, T. Fueno, K. Yamaguchi, Theor. Chim. Acta 48 (1978) 175. [2] K. Takatsuka, T. Fueno, J. Chem. Phys. 69 (1978) 661, and references cited therein. [3] R.C. Bochicchio, J. Mol. Struct. (Theochem) 429 (1998) 229. [4] V.N. Staroverov, E.R. Davidson, Chem. Phys. Lett. 340 (2001) 142, and references cited therein. [5] L. Lain, A. Torre, R.C. Bochicchio, R. Ponec, Chem. Phys. Lett. 346 (2001) 283. [6] R.C. Bochicchio, L. Lain, A. Torre, Chem. Phys. Lett. 375 (2003) 45. [7] M. Head-Gordon, Chem. Phys. Lett. 372 (2003) 508. [8] R.C. Bochicchio, J. Mol. Struct. (Theochem) 228 (1991) 227, and references cited therein. [9] H.F. Reale, R.C. Bochicchio, J. Phys. B 24 (1991) 2937, and references cited therein. [10] P. Ziesche, J. Mol. Struct. (Theochem) 527 (2000) 35.
Comment on ÔCharacterizing unpaired electrons from one-particle density matrixÕ [M. Head-Gordon, Chem. Phys. Lett. 372 (2003) 508–511] R.C. Bochicchio
a,* ,
A. Torre b, L. Lain
b
a
b
Departamento de Fısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pab. 1 – Ciudad Universitaria, Buenos Aires 1428, Argentina Departamento de Quımica Fısica, Facultad de Ciencias, Universidad del Paıs Vasco, Apdo. 644, Bilbao E-48080, Spain Received 9 June 2003; in final form 8 September 2003 Published online: 7 October 2003
Abstract A recent theoretical study published in this journal has proposed a new definition of the unpaired electron density in molecular systems. This comment attempts to clarify the difference between the concept reported there and that of ÔeffectivelyÕ unpaired electron density. Ó 2003 Elsevier B.V. All rights reserved.
Since the early introduction by Takatsuka et al. [1] and Takatsuka and Fueno [2] the effectively unpaired electron density has been subject of several studies [3–6]. However, there are some misunderstandings about this matrix, may be because of its name ‘effectively’ unpaired electron, as could be inferred from [7]. This report concludes that a contradiction would arise because in some cases the trace of the effectively unpaired electron density matrix Nu exceeds the total number of electrons N . If this interpretation were right, the effectively unpaired electron matrix u should not be as useful as was considered in the near past. In this Comment
we attempt to clarify the physical meaning of this matrix and the boundary values of its trace. For such a goal let us introduce the natural definition for u matrix that expresses the lack of idempotency of the spin-free 1-particle density matrix 1 D (1-RDM) [1–6] as X 1 i1 k uij ¼ 21 Dij Dk Dj ð1Þ k
and so trðuÞ ¼ 2N trð1 D2 Þ because trð1 DÞ ¼ N . In a more explicit manner, it can also be recognized as part of the master equation (particle-hole conservation equation), Þ; 2N ¼ trð1 D1 DÞ þ trð1 D1 D
*
Corresponding author. Fax: 54-11-45763357. E-mail address: [email protected] (R.C. Bochicchio).
ð2Þ
where the first and second terms of the RHS of Eq. (2) are the statistical averages of the number of
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.09.035
R.C. Bochicchio et al. / Chemical Physics Letters 380 (2003) 486–487
particles [8] and holes in the system [3], respectively; the last one is related to the u matrix by Þ: Nu trðuÞ ¼ trð1 D1 D
ð3Þ
Eq. (2) is a consequence of the hole-particle nonseparability in many body systems at correlated level showing that the boundary values for Nu are, 0 6 Nu 6 2N . In order to proceed to clarify the physical meaning of the u matrix and its boundary values we will make use of the structure of this matrix which can be expressed by 2
u ¼ ð1 Dspin Þ D; 1
spin
ð4Þ
where D is the 1-particle spin density matrix or simply the spin density matrix, expressed as the difference 1 Dspin ¼ 1 Da 1 Db , where 1 Dr ; r ¼ a; b stands for the spin up (a) and spin down ðbÞ one particle density matrices, respectively [6]. D ¼ Da þ Db collects the total deviation from idempotency as contributions of the spin up and the spin down clouds, respectively, defined by ð1 Dr Þ2 ¼ 1 r D þ 12Dr [6]. It can be noted that the natural matrix for describing the net unpaired particles density is the spin density matrix as has been well established [9] and not the u matrix, i.e., trð1 Dspin Þ ¼ trð1 Da Þ trð1 Db Þ ¼ N a N b where N a and N b are the number of a–electrons and b–electrons in the system, respectively. Thus, there is not any contradiction between the physical meaning of u matrix and its boundary values. However, in order to complete the analysis, let us inspect the values for Nu as a consequence of such structure. The upper value Nu ¼ 2N stands for a configuration in which all orbital occupation numbers are fni 1; i ¼ 1; . . . ; Kg where K is the size of the basis set, which indicates a highly ‘‘diluted’’ state or having a great amount of holes in the configuration, supporting an unphysical situation; the lower bound, Nu ¼ 0, is the most ‘‘condensed’’ electron configuration, i.e., the first N2 orbitals are doubly occupied while the remaining ones are vacant. These results seem to indicate that the upper bound describes a very disordered configuration while the lower one describes an ordered one. Another important value for Nu is when Nu N . A simple calculation from Eq. (4) shows that this condition is fulfilled by states having the highest spin projection, i.e., Sz ¼ N2 . This case at Hartree–Fock (Restricted
487
Open Shell HF-ROHF) level of description results in u ¼ 1 Da 1 D thus Nu ¼ N . At correlated level Nu ¼ trð1 Da Þ 12trðDa Þ ¼ N 12trðDa Þ which is also an improbable physical situation except in a simple system like hydrogen molecule. These results show that Nu increases as long as correlation and even delocalization are present in the system [10]. It can be noted that values Nu > N are not consistent for molecular systems even in nonequilibrium points defining a chemical rearrangement on the potential hypersurface because atoms preserve their fundamental electronic configuration during a chemical reaction and thus ‘‘extreme’’ states are not involved in such transformations. Therefore, u matrix so defined is a measure of the ‘‘deformation’’ of the ordered population when passing from Hartree– Fock reference states to correlated states in which orbitals are not double occupied. Because such occupations are not observable quantities the physical meaning of this matrix is related to valence concepts through the hole populations as demonstrated in [6]. Singlet states at correlated level show Nu 0 but not null which indicates an nonuniform orbital occupation due to correlation effects. Thus u cannot be considered as an net unpaired electron density and it can be interpreted as local space unpaired density (cf. discussion in [3]) or, as widely used in the literature, ‘‘effectively’’ unpaired electron density matrix.
References [1] K. Takatsuka, T. Fueno, K. Yamaguchi, Theor. Chim. Acta 48 (1978) 175. [2] K. Takatsuka, T. Fueno, J. Chem. Phys. 69 (1978) 661, and references cited therein. [3] R.C. Bochicchio, J. Mol. Struct. (Theochem) 429 (1998) 229. [4] V.N. Staroverov, E.R. Davidson, Chem. Phys. Lett. 340 (2001) 142, and references cited therein. [5] L. Lain, A. Torre, R.C. Bochicchio, R. Ponec, Chem. Phys. Lett. 346 (2001) 283. [6] R.C. Bochicchio, L. Lain, A. Torre, Chem. Phys. Lett. 375 (2003) 45. [7] M. Head-Gordon, Chem. Phys. Lett. 372 (2003) 508. [8] R.C. Bochicchio, J. Mol. Struct. (Theochem) 228 (1991) 227, and references cited therein. [9] H.F. Reale, R.C. Bochicchio, J. Phys. B 24 (1991) 2937, and references cited therein. [10] P. Ziesche, J. Mol. Struct. (Theochem) 527 (2000) 35.