Legendre transforms

A Legendre transform of a state function is a linear change of one or more of the independent variables made by subtracting products of conjugate variables. 

To understand how this works, consider a state function / whose total differential is given by 

For the first example of a Legendre transform, we define a new state function /i by subtracting the product of the conjugate variables a and x  

Equation F.4.4 gives the total differential of f with a, y, and z as the independent variables. The functions x and a have switched places as independent variables. What we did in order to let a replace x as an independent variable was to subtract from / the product of the conjugate variables a and x. 

Because the right side of Eq. F.4.4 is an expression for the total differential of the state function f, we can use the expression to identify the coefficients as partial derivatives of /i with respect to the new set of independent variables  

We derive the transform first using the simple algebraic approach of Boas (1966, p. 159). The total differential of a function / = f x, y) is written 

Notice that the transformation merely replaces the original variable y in f x, y) by its partial derivative df /dy to give the new function (x, df /dy), which is 

For many people, it is helpful to view this solution geometrically (Callen, 1960, p. 90). To state the problem in a different way then, given the function 

That is, to form the Legendre Transform of a function, subtract from the original function the products of each variable to be changed and the derivative of the function with respect to that variable. After that, one can proceed to tidy up by eliminating y in the new function by differentiating. Thus in the case of y = y(x), 

Frankly, as far as its use in thermodynamics is concerned, we could stop right there. However, to demonstrate that we have defined a function g that really has as its independent variables y and (dx/dz), which is not exactly obvious at this stage, we can substitute (y — 3z ) for x and —6z for (dx/dz), giving 

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By differentiating the defining equations for H, A and G and combining the results with equation (A2.T25) and equation (A2.T27) for dU and U (which are repeated here) one obtains general expressions for the differentials dH, dA, dG and others. One differentiates the defined quantities on the left-hand side of equation (A2.1.34), equation (A2.1.35), equation (A2.1.36), equation (A2.1.37), equation (A2.1.38) and equation (A2.1.39) and then substitutes die right-hand side of equation (A2.1.33) to obtain the appropriate differential. These are examples of Legendre transformations. ... 

Equation 54 implies that U is a function of S and P, a choice of variables that is not always convenient. Alternative fundamental property relations may be formulated in which other pairs of variables appear. They are found systematically through Legendre transformations (1,2), which lead to the following definitions for the enthalpy, H, Hehnholt2 energy,, and Gibbs energy, G ... 

To recast the thermodynamic description in terms of independent variables that can be controlled in actual laboratory experiments (i.e., T, /i, and the set of strains or their conjugate stresses), it is sensible to introduce certain auxiliary thermodynamic potentials via Legendre transformations. This chapter is primarily concerned with... 

A familiar example of Legendre transformation is the relationship that exists between the Lagrangian and Hamiltonian functions of classical mechanics [17]. In thermodynamics the simplest application is to the internal energy function for constant mole number U(S, V), with the differentials... 

In similar vein the enthalpy H = U(S, P, Nj) is the partial Legendre transform of U that replaces the volume by pressure as an independent variable. Recall that dU/dV — —P, so that the appropriate Legendre transform is... 

The enthalpy is useful in considering isentropic and isobaric processes, but often it becomes necessary to rather deal with isothermal and isobaric processes. In such case one needs a thermodynamic function of T and P alone, defining the Gibbs potential G = U(T, P, Nj) as the Legendre transform of U that replaces entropy by temperature and volume by pressure. This transform is equivalent to a partial Legendre transform of the enthalpy,... 

A Legendre transformation to switch between mole number and chemical potential may also be performed, for instance to produce a potential U(T, V, p) for a one-component system as... 

The potential function of intensive variables only, U(T, P, pj) corresponds to the full Legendre transformation... 

Legendre transformation does not affect the essential nature of a function and all of the different potentials defined above still describe the internal energy, not only in terms of different independent variables, but also on the basis of different zero levels. In terms of Euler s equation (2) the internal energy consists of three components... 

As in energy representation the fundamental thermodynamic equation in entropy representation (3) may also be subjected to Legendre transformation to generate a series of characteristic functions designated as Massieu-Planck (MP) functions, m. The index m denotes the number of intensive parameters introduced as independent variables, i.e. 

The statistical partition functions are seen to be related by Laplace transformation in the same way that thermodynamic potentials are related by Legendre transformation. It is conjectured that the Laplace transformation of the statistical partition functions reduces asymptotically to the Legendre transformation of MP in the limit of infinitely large systems. 

In order to focus on the driving force for phase transitions induced by a magnetic field it is advantageous to use the magnetic flux density as an intensive variable. This can be achieved through what is called a Legendre transform [12], A transformed Helmholtz energy is defined as... 

In 1982, Nalewajski and Parr took the thermodynamic analogy to its logical conclusion by extending the Legendre-transform structure of classical thermodynamics to DFT [8]. One of their results was the Maxwell relation for Equation 18.6,... 

But there was no significant further development of the ideas set forth by Parr and Bartolotti until 1994, when one of the present authors (A.C.) realized that the shape function provided the key to resolve a mathematical difficulty that is inherent in the different Legendre-transform representations of chemical DFT [20]. 

The use of the electron density and the number of electrons as a set of independent variables, in contrast to the canonical set, namely, the external potential and the number of electrons, is based on a series of papers by Nalewajski [21,22]. A.C. realized that this choice is problematic because one cannot change the number of electrons while the electron density remains constant. After several attempts, he found that the energy per particle possesses the convexity properties that are required by the Legendre transformations. When the Legendre transform was performed on the energy per particle, the shape function immediately appeared as the conjugate variable to the external potential, so that the electron density was split into two pieces that can be varied independent the number of electrons and their distribution in space. 

Just as in classical statistical mechanics, the different pictures of electronic changes are related by Legendre transforms. The state function for closed systems in the electron-following picture is just the electronic ground-state energy, /i v AT The total differential for the energy provides reactivity indicators for describing how various perturbations stabilize or destabilize the system,... 

Moving from the closed-electron-following picture (canonical ensemble) to the open-electron-following picture (grand canonical ensemble) is done by Legendre transform, generating the new state function ... 

The closed-electron-preceding picture (isomorphic ensemble) requires a Legendre transform to eliminate the external potential as a variable,... 

Equation 19.17 is not the original way the N, p(r) ambiguity was resolved [20]. As mentioned previously, the original paper on the shape function in the isomorphic representation performed the Legendre transform on the energy per particle. This gives an intensive, per electron, state function [20] ... 

Another example is provided by the minimum energy coordinates (MECs) of the compliant approach in CSA (Nalewajski, 1995 Nalewajski and Korchowiec, 1997 Nalewajski and Michalak, 1995,1996,1998 Nalewajski etal., 1996), in the spirit of the related treatment of nuclear vibrations (Decius, 1963 Jones and Ryan, 1970 Swanson, 1976 Swanson and Satija, 1977). They all allow one to diagnose the molecular electronic and geometrical responses to hypothetical electronic or nuclear displacements (perturbations). The thermodynamical Legendre-transformed approach (Nalewajski, 1995, 1999, 2000, 2002b, 2006a,b Nalewajski and Korchowiec, 1997 Nalewajski and Sikora, 2000 Nalewajski et al., 1996, 2008) provides a versatile theoretical framework for describing diverse equilibrium states of molecules in different chemical environments. 

The inverse of H determines the geometric compliance matrix (Nalewajski, 1993, 1995, 1997, 1999, 2000, 2002b, 2006a,b Nalewajski and Korchowiec, 1997 Nalewajski et al., 1996, 2008) describing the open system in the Qi,F)-representation. The relevant thermodynamic potential is defined by the total Legendre transform of the system BO potential, which replaces the state-parameters (N, Q) with their energy conjugates (/a, F), respectively ... 

Let us now turn to the mixed, partly inverted (N, F)-representation describing the geometrically relaxed, but externally closed molecular system. The relevant thermodynamic potential is now defined by the partial Legendre transformation of W(N, Q) which replaces Q by F in the list of the system parameters of state ... 

Finally, the remaining (/a, Q) representation describing the equilibrium state of an externally open molecular system with the frozen nuclear framework is examined. The relevant partial Legendre transform of the total electronic energy, which replaces N by /a in the list of independent state-parameters, defines the BO grand-potential ... 

The electronic-nuclear coupling in molecules is also detected in the other partial Legendre-transformed representation H(/r, Q), which defines the combined Hessian G of Equation 30.27. Its first diagonal derivative,... 

Here (N, F) (see Equation 30.17) stands for the Legendre transform of the BO potential energy surface W(N, Q), in which the nuclear-position coordinates Q are replaced by the corresponding forces F in the list of the parameters of state. Indeed, for the fixed number of electrons N,... 

Obviously, the partly inverted Legendre-transformed representations for reactive systems would similarly generate descriptors of the partially relaxed (electronically or geometrically) reactive systems. 

This definition is an example of a Legendre transform, as discussed by some authors, such as R. A. Alberty, Pure Appl. Chem. 73, 1349 (2001). 

Equation A2.19 is one of the possible ways to switch from a couple of variables to another conjugate couple. Most transforms adopted in the thermodynamics of systems with two independent variables are based on equation A2.19 and its modifications (Legendre transforms, cf section 2.4). 

P. W. Ayers, S. Golden, and M. Levy, Generalizations of the Hohenberg—Kohn theorem I. Legendre transform constructions of variational principles for density matrices and electron distribution functions. J. Chem. Phys. 124, 054101 (2006). 

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