### Thermodynamic Dual Descriptor

FRANCO-PÉREZ Marco1,*(),GÁZQUEZ José L.1,AYERS Paul W.2,VELA Alberto3

1. 1 Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Ciudad de México, 09340, México
2 Department of Chemistry, McMaster University, Hamilton, Ontario, L8S 4M1, Canada
3 Departamento de Química, Centro de Investigación y de Estudios Avanzados, Av. Instituto Politécnico Nacional 2508, Ciudad de México, 07360, México
• 收稿日期:2017-11-16 发布日期:2018-03-20
• 通讯作者: FRANCO-PÉREZ Marco E-mail:qimfranco@hotmail.com

### Thermodynamic Dual Descriptor

Marco FRANCO-PÉREZ1,*(),José L. GÁZQUEZ1,W. AYERS Paul2,Alberto VELA3

1. 1 Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Ciudad de México, 09340, México
2 Department of Chemistry, McMaster University, Hamilton, Ontario, L8S 4M1, Canada
3 Departamento de Química, Centro de Investigación y de Estudios Avanzados, Av. Instituto Politécnico Nacional 2508, Ciudad de México, 07360, México
• Received:2017-11-16 Published:2018-03-20
• Contact: Marco FRANCO-PÉREZ E-mail:qimfranco@hotmail.com

A new definition of the dual descriptor, namely, the thermodynamic dual descriptor, is developed within the grand canonical potential formalism. This new definition is formulated to describe the same physical phenomenon as the original definition proposed by Morell, Grand, and Toro-Labbé (J. Phys. Chem. A 2005, 109, 205), which is characterized by a second-order response of the electron density towards an electron flux. To formulate the new definition, we performed two successive partial derivatives of the average electron density, one with respect to the average number of electrons, and the other with respect to the chemical potential of the electron reservoir. When the derivative is expressed in terms of the three-state ensemble model, in the regime of low temperatures up to temperatures of chemical interest, one finds that the thermodynamic dual descriptor can be expressed as ∆fT(r) = (β/2)C[f+(r)-f-(r)], where β = 1/kBT, C is a global quantity that depends on the temperature and global electronic properties of the molecule (the first ionization potential and the electron affinity), C = 1 for systems with zero fractional charge, and C = Cω > 0 (albeit very close to zero) for systems with nonzero fractional charge, , and f+(r) and f-(r) are the nucleophilic and electrophilic Fukui functions, respectively. The quantity within the square brackets is the original definition of the dual descriptor. As the local terms (the ones containing regioselectivity information) are equal to those of the dual descriptor, ∆fT(r) has the same regioselectivity information, multiplied by the global quantity (β/2)C. This implies that the regioselectivity information contained in the original dual descriptor is preserved at all temperatures different from zero, and for any value of C > 0. One of the most important features of this new definition is that it avoids the undesired Dirac delta behavior observed when the second order partial derivative of the average density is taken with respect to the average number of electrons, using the exact density dependence of the average number of electrons.

Abstract:

A new definition of the dual descriptor, namely, the thermodynamic dual descriptor, is developed within the grand canonical potential formalism. This new definition is formulated to describe the same physical phenomenon as the original definition proposed by Morell, Grand, and Toro-Labbé (J. Phys. Chem. A 2005, 109, 205), which is characterized by a second-order response of the electron density towards an electron flux. To formulate the new definition, we performed two successive partial derivatives of the average electron density, one with respect to the average number of electrons, and the other with respect to the chemical potential of the electron reservoir. When the derivative is expressed in terms of the three-state ensemble model, in the regime of low temperatures up to temperatures of chemical interest, one finds that the thermodynamic dual descriptor can be expressed as ∆fT(r) = (β/2)C[f+(r)-f-(r)], where β = 1/kBT, C is a global quantity that depends on the temperature and global electronic properties of the molecule (the first ionization potential and the electron affinity), C = 1 for systems with zero fractional charge, and C = Cω > 0 (albeit very close to zero) for systems with nonzero fractional charge, , and f+(r) and f-(r) are the nucleophilic and electrophilic Fukui functions, respectively. The quantity within the square brackets is the original definition of the dual descriptor. As the local terms (the ones containing regioselectivity information) are equal to those of the dual descriptor, ∆fT(r) has the same regioselectivity information, multiplied by the global quantity (β/2)C. This implies that the regioselectivity information contained in the original dual descriptor is preserved at all temperatures different from zero, and for any value of C > 0. One of the most important features of this new definition is that it avoids the undesired Dirac delta behavior observed when the second order partial derivative of the average density is taken with respect to the average number of electrons, using the exact density dependence of the average number of electrons.