In what follows we shall build up upon the basic DFT variational equation, which summarizes the second Hohenberg-Kohn theorem, and develop the equivalent (coupled) “thermodynamic”-like principles, which simultaneously involve the electronic energy and the Kullback-Leibler entropy-deficiency (information-penalty) term (Nalewajski, 2005d). Hongrui Wang, Hongwei Liu, Leixin Cai, Caixia Wang, Qiang Lv. (c) The potential energy curves of the variationally optimized closed‐shell states and spin‐adapted open‐shell states. The ΔS[ρ’| ρ] “intensive” conjugate of the “extensive” variable ρ is then defined by the functional derivative. By continuing you agree to the use of cookies. This allows the calculation of analytic gradients, a necessity for efﬁcient molecular dynamics simulations. A 14, 36 (1976)] seeks the local potential that is variationally optimized to best approximate the Hartree-Fock exchange operator. Finally, the coefficients c(k) are variationally optimized to obtain the best energy, using a generalization of an algorithm originally developed for the Coulomb potential [J. M. Perez-Jorda, Phys. Lett. In this work, we use slater-type orbitals (STOs) as basis functions to construct exciton and trion wavefunctions in two dimensions. Your Mendeley pairing has expired. Get article recommendations from ACS based on references in your Mendeley library. (10.2.10) and (10.2.15) is given by the system chemical potential of Eq. In what follows we shall demonstrate that a non-equalized (local) information temperature is required in the “thermodynamic” principle determining the non-equilibrium electron distribution ρ’: ∆ρ = ρ’− ρ ≠ 0. In this study, it is shown that the exciton energy levels can be calculated accurately by a variationally optimized pro-cedure. which is seen to be equalized throughout the physical space for ρ’(r)= ρ(r). A variational approach to nucleation simulation. Citations are the number of other articles citing this article, calculated by Crossref and updated daily. the Altmetric Attention Score and how the score is calculated. The associated Euler equation [see Eqs. We use cookies to help provide and enhance our service and tailor content and ads. Information about how to use the RightsLink permission system can be found at James McCarty, Omar Valsson, and Michele Parrinello . The kinetic energy operator depends parametrically on the gauge origin O, an arbitrary point in space where the field contribution to the operator vanishes. Physical Review Letters 2015 , 115 (7) DOI: 10.1103/PhysRevLett.115.070601. Here, we present a simple approach to calculate the gauge correction necessary to eliminate this inconsistency. Therefore, the trial potentials in this constrained search can only differ by a constant μ = μℜ. Neural networks-based variationally enhanced sampling. The DFT variational principle is thus interpreted as being equivalent to the “thermodynamic” principle of the IT approach. 3. (3) The RVM is used to drastically sparsify the data set without changing the learned function. Coarse graining from variationally enhanced sampling applied to the Ginzburg–Landau model. (2) The weights of each configuration are optimized variationally, reasonably describing the desired function. The most natural candidate for such a penalty function is the directed-divergence (entropy-deficiency) functional of Kullback and Leibler (see Section 3.3). Additionally, the asymptote behavior of STOs is known to be fitted to wavefunctions of Coulombian few-particle systems in regimes near two … 90, 317 (1953)] and Talman and Shadwick [Phys. http://pubs.acs.org/page/copyright/permissions.html, https://doi.org/10.1021/acs.jpclett.0c00497, https://doi.org/10.1021/acs.jpclett.9b03356, https://doi.org/10.1021/acs.jpclett.8b03005, https://doi.org/10.1021/acs.jpclett.7b01770, https://doi.org/10.1007/978-3-319-44677-6_50, https://doi.org/10.1103/PhysRevLett.122.050601, https://doi.org/10.1007/978-1-4939-9608-7_21, https://doi.org/10.3390/computation6010015, https://doi.org/10.1007/978-3-319-42913-7_50-1, https://doi.org/10.1186/s12859-017-1733-6, https://doi.org/10.1103/PhysRevLett.119.015701, https://doi.org/10.1140/epjst/e2016-60094-x, https://doi.org/10.1146/annurev-physchem-040215-112229, https://doi.org/10.1103/PhysRevLett.115.070601. Files available from the ACS website may be downloaded for personal use only. The radial function of basis orbitals can be variationally optimized using the orbital optimization method [].As an illustration of the orbital optimization, let us explain it using a methane molecule of which input file is 'Methane_OO.dat'. not otherwise permitted to reproduce, republish, redistribute, or sell any Supporting Information A generalized RPA method extracts excitation energies from a correlated ground state second order density matrix. Additionally, the asymptote behavior of STOs is known to be tted to wavefunctions of Coulombian few-particle systems in regimes near two-particle coalescence In this approach, to which both the variational functional Ev[ρ], for a trial ρ, and the ground-state functional E¯uρu=E¯u belong, the electron density represents the dependent state-variable. Find more information on the Altmetric Attention Score and how the score is calculated. As we have also emphasized in Section 1.5, this DFT Euler equation is exactly equivalent to the associated Schrödinger principle (1.5.7), with respect to the trial wave-functions of N electrons, Ψ(N), for the minimum expectation value of the electronic energy subject to the constraint of the wave-function normalization, N [Ψ] = 〈Ψ | Ψ〉 = 1, enforced by the energy Lagrange multiplier E[N, v] = (∂Ev/∂N])v: One then defines the deviation of the current (variational) energy from the exact ground-state energy level for a given trial density ρ’. Alternatively, this density constraint can be imposed indirectly, in a thermodynamic-like manner, by using an appropriate information-penalty term, e.g., the cross-entropy in the current density ρ’ relative to ρ, representing the information distance between the two compared electron distributions. The method finds a variationally optimized function of a contour that characterizes the system. Alternatively, the density constraint ρ’(r) = ρ(r) can be imposed through the local “thermodynamic” constraint, by fixing the the entropy-deficiency density of [Eq. Copyright © 2020 Elsevier B.V. or its licensors or contributors. Patrick Shaffer, Omar Valsson, Michele Parrinello. It was shown in [l] that the variational problem leads to a linear integral equation (a Fredholm equation of the first kind) for Additional convergence results. Hence, one obtains ϕ≈12m0c(σ⋅π)ψ≈υ2cψ, therefore the component ϕ is for small v much smaller than the component ψ, which justifies the terms “small” and “large” components.26, Roman F. Nalewajski, in Information Theory of Molecular Systems, 2006. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. & Account Managers, For We propose a simple yet effective iterative scheme that allows us to employ the well-tempered distribution as a target distribution for the collective variables in our recently introduced variational approach to enhanced sampling and free energy calculations [Valsson and Parrinello Phys. Using the Maximum Entropy Principle to Combine Simulations and Solution Experiments. Spectrosc. Atomistic-based simulations are one of the most widely used tools in contemporary science. This article is cited by Luigi Bonati, Yue-Yu Zhang, Michele Parrinello. Information. Pablo M. Piaggi, Omar Valsson, Michele Parrinello. Indeed, in the Hohenberg-Kohn [Eq. In the latter the electronic chemical potential plays the role of the information “temperature”, which enforces the ground-state entropy-deficiency constraint. (1.5.22), the universal density functional F[ρ] of the HR (in the EP perspective), can be alternatively regarded as the associated functional of the relative external potential F¯[u] of the SR (in the EF perspective): The corresponding density constrained search for the external potential matching the given (v-representable) density ρ reads (see, e.g., Lieb, 1982; Nalewajski and Parr, 1982; Lieb, 1983; Nalewajski and Korchowiec, 1997; Colonna and Savin, 1999): In this extremum principle one searches over external potentials v’→ρ, which give rise to for the specified ground-state density ρ, and determines the maximum/suprimum of the Legendre transform (see the next chapter) of the system energy, which replaces the external potential v’ by the ground-state density ρ, in the list of the system independent state-parameters. Zhongji Pu, Mengdi Zhao, Yue Zhang, Wenhui Sun. Journal of Chemical Theory and Computation. (1.5.14), for the variational energy Ev[ρ], in which the external potential is fixed (not related to the trial density ρ). (1.5.21)], which is equalized throughout the space: In a more compact form of Eq. James McCarty, Omar Valsson, Pratyush Tiwary, Michele Parrinello. Find more information about Crossref citation counts. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.5b00076. Please note: If you switch to a different device, you may be asked to login again with only your ACS ID. This will give a variationally optimized approximation to the ground state energy of Hˆ (plus a concomitant approximation to the corresponding ground state energy eigenfunction). June 2015; Physical Review Letters 115(7) ... V ariationally Optimized F ree Energy Floo ding for Rate Calculation. Free energy barriers therefore cannot be consistently computed from the FES. (10.2.2) can be automatically satisfied, when the variational procedure guarantees that the optimum density ρopt = [N, v]. Therefore, such a softness “picture” adopts the EF perspective of Section 1.2. 2014, 113, 090601]. (10.2.2) and (10.2.6) become identical. Reference free energy surfaces used for the ε error metric. to study exciton energy levels in 2D materials[32, 33]. In the variational diabatic configuration (VDC) method, the energies of the individual diabatic states are variationally minimized. Samuel Alexander Jobbins, Salah Eddine Boulfelfel, Stefano Leoni. Biomolecular conformational changes and ligand binding: from kinetics to thermodynamics. Highlights We consider the problem of computing molecular excitation spectra from ground state density matrices. Calculation of phase diagrams in the multithermal-multibaric ensemble. First, the wave function for the monomeric system is variationally optimized using standard X-Pol method, as opposed to the iterative update procedure adopted in FMO. because, empirically, the energy does converge with respect to enlargement of the variational space.24–26 Thus, while the variationally optimized energy in approximate DFT may be higher or lower than the exact energy, the model exchange– correlation hole is at least associated with a ﬁnite energy in the complete-basis limit. In the “thermodynamic” treatment of molecular systems (Ghosh, et al., 1984; Ghosh and Berkowitz, 1985; Nagy and Parr, 1994; Nalewajski and Parr, 2001; Nalewajski 2002c, 2003a,b, 2004a, 2005d) one aims at determining changes in the electronic structure accompanying the displacement from one equilibrium (ground-state) density, ρ1 = ρ[N1, v1], to another, ρ2 = ρ[N2, v2]. Multithermal-Multibaric Molecular Simulations from a Variational Principle. It introduces the entropic “penalty” for the trial density deviating from the true ground-state density ρ, and combines the system electronic energy and its average missing-information functional. Appealing to the linear variation of parameters version of the Variational Principle, we need to solve the eigenvalue/eigenvector problem: Enhancing Important Fluctuations: Rare Events and Metadynamics from a Conceptual Viewpoint. Conformational Entropy as Collective Variable for Proteins. Phase equilibrium of liquid water and hexagonal ice from enhanced sampling molecular dynamics simulations. Rev. The horizontal displacement, from one equilibrium (v-representable) electron density to another, gives rise to the associated change in the generalized density functional for the ground-state energy. so that that the minimum-energy principles of Eqs. Reviewers, Librarians This is because the ground-state Euler equation (1.5.22) shows that the density and relative external potential are unique functionals of each other. We apply this method to exact and variationally optimized second order density matrices. Since u = u[ρ] and ρ = ρ[u], by Eq. You’ve supercharged your research process with ACS and Mendeley! You have to login with your ACS ID befor you can login with your Mendeley account. (10.2.9) and (10.2.13). Via this method, exciton and trion energy levels and wavefunctions can be calculated efficiently with fairly accurate results. Andrea Cesari, Sabine Reißer, Giovanni Bussi. (10.2.8)], ∆s(r) = 0, using the local Lagrange multiplier τ(r): The latter is now defined by the functional derivative. Here, we show that by combining a variational approach with deep learning, much progress can be made in extending the … (10.2.3). The results observed in this paper indicate that the well-tempered distribution is in most cases the preferred and recommended choice for the target distribution in the variational approach. This is also the case in the Levy (1979) constrained search construction of the universal functional F[ρ] of Eqs. Unfortunately not every imple-mentation of molecular dynamics obeys eq. Since this search is performed for the fixed (ground-state) density, it also implies the fixed value of the system electronic energy. Yong Wang, João Miguel Martins, Kresten Lindorff-Larsen. Clearly, the subsidiary condition of the required normalization of the optimum electron density in the DFT variational principle of Eq. One should distinguish between (a) Noether type energy-momentum tensors, that arises when the underlying variational functionals are invariant with respect to a given Lie group, and (b), Elementary Methods of Molecular Quantum Mechanics, Elements of the Information–Distance Thermodynamics, Journal of Mathematical Analysis and Applications, Computer Methods in Applied Mechanics and Engineering. (1.5.22) this Euler equation involves the relative external potential, u(r) = v(r) − μ = − δF[ρ]/δρ(r) = u [ρ; r], the unique functional of the ground-state density. which identically vanishes for ρ’(r)= ρ(r). (4) Configurations with high local energy variance are added to … We call such shifts in the system electronic structure the “horizontal” displacements (Nalewajski, 1999; Nalewajski and Parr, 2001) on the ground-state density ρ[N, v] = ρ and energy E[N, v] = Ev[ρ] = Egs[ρ] “surfaces”. (1). where F˜aρ denotes the additive part of F˜aρ, is carried out for the fixed densities of the remaining subsystems {ρβ≠ α}. A method to calculate analytical energy gradients for multiconfiguration self-consistent field (MCSCF) wave functions with frozen core orbitals is presented. Rev. Therefore, by analogy to the maximum principle of the thermodynamic entropy for constant internal energy in the ordinary phenomenological thermodynamics, this DFT minimum principle can be regarded as being also “entropic” in character. Effect of ethanol on insulin dimer dissociation. This is done by variationally optimizing eigenstates of integrable Richardson-Gaudin models, where the toolbox of integrability allows for an efficient evaluation and minimization of the energy functional. Here, the nonlinear activation function is taken to be a rectiﬁed linear unit. (1.5.21), which characterizes the molecular system as a whole: where the unit row vector 1 = 1, 1, …) and the vector N ≡ {Nα = ∫ ρα(r) dr ≡ N[ρα]} groups the average numbers of electrons in atomic subsystems. This constrained-search construction can be generalized to any trial density ρ’ (Lieb, 1983; Colonna and Savin, 1999): As argued elsewhere (Nalewajski and Parr, 2001), the AIM division problem of the fixed molecular density ρ also represents a search for the optimum effective external potentials of atomic subsystems veff = {vαeff}: Here, the partial differentiation with respect to ρα(r) of the non-additive part F˜nρ of the total Hohenberg–Kohn–Levy functional in the AIM resolution, Fρ=F˜ρ. Analyzing and Biasing Simulations with PLUMED. Variationally Optimized Free Energy Flooding for Rate Calculation. Tarak Karmakar, Pablo M. Piaggi, Claudio Perego. Librarians & Account Managers. This global temperature parameter has also been shown to represent the equalized value of the local information temperature, τ(r) = μ(r) characterizing the specified infinitesimal volume element of the ground-state (equilibrium) distribution of electrons. Rethinking Metadynamics: From Bias Potentials to Probability Distributions. Enhanced, targeted sampling of high-dimensional free-energy landscapes using variationally enhanced sampling, with an application to chignolin. The performance of the scheme is evaluated for the three-dimensional free energy surface of alanine tetrapeptide where the convergence can be rather poor when employing the uniform target distribution. In this paper, the detailed derivations of the Fock … At the solution point this variational principle yields the optimum external potential v = v[ρ], which identifies the specified (v-representable) density as its ground-state, equilibrium distribution. We recall that the Hohenberg-Kohn (1964) principle of Eq. Rev. In the consistent diabatic configuration (CDC) method, both the configuration coefficients and orbital coefficients are simultaneously optimized to minimize the adiabatic ground-state energy in VBSCF calculations. 1. The European Physical Journal Special Topics. Metashooting: a novel tool for free energy reconstruction from polymorphic phase transition mechanisms. Michele Invernizzi, Omar Valsson, Michele Parrinello. (10.2.3). π is of the order of momentum mv, and for the small velocities of the order of m0v. (10.2.3) and (10.2.9)] then gives: Hence, this alternative, local formulation also identifies the equilibrium local chemical potential for the ground-state density, μ(r) = μ, as the local (equalized) value of the information temperature: τ(r) = τ. system. spin-orbitals are variationally optimized to be diﬀerent if this lowers the computed energy (which is a standard way of performing practical KS-DFT calculations on open-shell systems This coupled (“thermodynamic”) principle identifies τ as the system global information “temperature”: It measures the linear response of the system electronic energy. Evaluation of computational options for updating the well-tempered target distribution. Rev. However, in the presence of kinetic bottlenecks, their power is severely curtailed. http://pubs.acs.org/page/copyright/permissions.html. which can thus be interpreted as the local information temperature. Spectral gap optimization of order parameters for sampling complex molecular systems. from the ACS website, either in whole or in part, in either machine-readable form or any other form Electronic Supporting Information files are available without a subscription to ACS Web Editions. Using the multi-objective optimization replica exchange Monte Carlo enhanced sampling method for protein–small molecule docking. Variationally optimized atomic orbitals for large-scale electronic structures ... optimized after cm,iaare determined with a set of ﬁxedaiaq. In order to mitigate this problem, many enhanced sampling techniques have been proposed. Over the course of the simulation, the Hamiltonian is changed continuously between the system of interest and a reference system for which the partition function has an analytic form. Puja Banerjee, Sayantan Mondal, Biman Bagchi. For example, this can be done directly, by imposing the local constraint ρ’(r) = ρ(r) multiplied by the corresponding local Lagrange multiplier λ(r) in a trivial Euler-Lagrange problem: One then identifies λ(r) as the local chemical potential of Eq. Discussion on the averaged update option for the well-tempered target distribution. 172, 265-274 (1995)], the electric dipole moment components were calculated directly as expectation values with the corresponding length operators … The Journal of Physical Chemistry Letters. These effective external potentials of the embedded bonded-atoms in a molecule are then related to their respective densities through the global-like ground-state Euler equation of DFT [Eqs. The same, equalized chemical potentials of AIM derived from the molecular ground-state density follow from the density functional for the electronic energy of the embedded atom α: where Ev[ρα] stands for the electronic energy of ρα alone, and Fn[ρ] represents the embedding energy due to the presence of electrons of the remaining subsystems (Nalewajski and Parr, 2001): We therefore conclude that the DFT description of the equilibrium states in molecules and their mutually-open subsystems are isomorphic, since the subsystem effective (relative) external potentials are related to their electron densities through the same “horizontal”-type Euler equation linking the complementary softness (EF) and hardness (EP) representations of the molecular electronic structure. Well-Tempered Variational Approach to Enhanced Sampling, Department of Chemistry and Applied Biosciences, ETH Zurich, c/o USI Campus, Via Giuseppe Buffi 13, CH-6900, Lugano, Ticino, Switzerland, Facoltà di Informatica, Instituto di Scienze Computationali, and National Center for Computational Design and Discovery of Novel Materials MARVEL, Università della Svizzera italiana (USI), Via Giuseppe Buffi 13, CH-6900, Lugano, Ticino, Switzerland. We therefore conclude that the information temperature in “thermodynamic” variational principles of Eqs. An approximate electronic wave function, consisting of 29026 basis functions for the entire system, is variationally optimized to give the minimum Born-Oppenheimer energy at every MD step; this allows the efficient evaluation of the required analytic forces for the dynamics. In this study, it is shown that the exciton energy levels can be calculated accurately by a variationally optimized procedure. Using the well-tempered target distribution on the other hand results in a significant improvement in convergence. in which the external potential changes with the electron density in such a way that the current electron distribution ρ matches v = v[ρ] as its ground-state (equilibrium) density. Together with the recently determined potential energy surface for the ground electronic state of HeH2+ [V. Spirko and W. P. Kraemer, J. Mol. Orbital optimization. variationally to minimize the ground state energy (or as dis- cussed below, a weighted average of low-lying term energies) calculated using the full non-relativistic Hamiltonian. The shielding constants of the STOs are treated as variational parameters. Hence the physical interpretation of the global information-temperature as the system chemical potential: This somewhat surprising identification is a direct consequence of Eqs. Users are where δρ(r) = δ [ρ’(r) − ρ(r)] ≡ δ∆ρ(r) and δN = ∫ δρ(r)dr, per unit displacement in the system global entropy deficiency: which amounts to the displacement in the system number of electrons. 32 publications. We are employing NNs since they are smooth interpolators. Optimized Energyが提供するサービスの例 Optimized Energyは他社にはまねできない オリジナルの電力契約を中心に、 さまざまなメニューを提供しています。 It should be recalled that ρ fixes the shape of v’ = v’[μ’, ρ], where μ’ denotes the system chemical potential attributed to an external reservoir. It should be emphasized that this density functional for the ground-state energy differs from the Hohenberg–Kohn functional of Eq. These electron distributions are uniquely determined by the two state-parameters determining the corresponding electronic Hamiltonians: the overall number of electrons Ni and the external potential due to the nuclei, vi(r), i = 1, 2. Please reconnect, Authors & and ; the full lines represent the variationally optimized linear combinations of these determinants. Ensemble-Based Molecular Simulation of Chemical Reactions under Vibrational Nonequilibrium. This density division problem is “vertical” in character (Nalewajski and Parr, 2000, 2001) being performed for the fixed molecular density. Dynamic Description of the Catalytic Cycle of Malate Enzyme: Stereoselective Recognition of Substrate, Chemical Reaction, and Ligand Release. so that δΔEv[ρ ’ |ρ] = δEv[ρ ’] and ΔEv[ρ|ρ] = 0. A Cannibalistic Approach to Grand Canonical Crystal Growth. The block-localized wavefunction at this stage is variationally optimized (eqn (6)), and the corresponding energy change is defined as the polarization … Ferruccio Palazzesi, Omar Valsson, and Michele Parrinello . Variationally Optimized Free-Energy Flooding for Rate Calculation. Variationally Optimized Free-Energy Flooding for Rate Calculation. Selecting the (electron) density as an independent state-variable, e.g., in the variational functional Eρ[u] , for a trial u, and the ground-state functional Eρ[u[ρ]] = E[ρ], gives the complementary Hardness Representation (HR), which adopts the EP perspective of Section 1.2. A generalization of the Feshbach-Rubinow method for variationally determining the ground-state energy of a multiparticle system is described and applied to the H - ion. Find more information about Crossref citation counts. Being based on the average value of the energy, the variational method privileges the space regions near to the nucleus, where the potential energy is larger (r small). Neural Network and Nearest Neighbor Algorithms for Enhancing Sampling of Molecular Dynamics. We present a variational method for approximating the ground state of spin models close to (Richardson-Gaudin) integrability. Second, the mutual polarizations in the dimeric terms are also variationally determined, whereas single-point energy calculations of the individual dimers embedded in a static monomer field are used in FMO. (1.5.20). parameters are optimized according to the variational principle of Eq. of the system electronic energy in the AIM resolution: Clearly, for the mutually open atomic subsystems, with no barriers preventing the flow of electrons between the bonded atoms, the AIM chemical potentials are equalized at the global chemical potential level of Eq. Enhancing Entropy and Enthalpy Fluctuations to Drive Crystallization in Atomistic Simulations. Orbital optimized unitary coupled cluster theory for quantum computer Wataru Mizukami, Kosuke Mitarai, Yuya O. Nakagawa, Takahiro Yamamoto, Tennin Yan, and Yu-ya Ohnishi Phys. B 58, 1230 (1998)]. Finite-time variational switching is an efficient method for obtaining converging upper and lower bounds to free energy changes by computer simulation. redistribute this material, requesters must process their own requests via the RightsLink permission Only for the exact ground-state density, satisfying the HK minimum principle of Eq. Use of variationally optimized wavefunctions can give poor results for operators different from H ˆ HC (like the dipole moment operator μ = e r , which takes large values far from the nucleus). Omar Valsson, Pratyush Tiwary, Michele Parrinello. We have indicated in the preceding equation that the energy functional E[ρ] can be alternatively interpreted as the functional of the relative external potential u(r), E¯u. (10.2.2) and the foregoing “thermodynamic” principle, of the minimum electronic energy for constant (vanishing) entropy-deficiency, are equivalent having both the system ground-state density as their solutions. (10.2.2)] and global “thermodynamic” [Eq. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/B9780080983561000102, URL: https://www.sciencedirect.com/science/article/pii/B9780444528339500164, URL: https://www.sciencedirect.com/science/article/pii/B9780444527783500220, URL: https://www.sciencedirect.com/science/article/pii/B9780444522276500041, URL: https://www.sciencedirect.com/science/article/pii/B9780444519665500748, URL: https://www.sciencedirect.com/science/article/pii/B9780444519665500657, The Finite Element Method (Second Edition), Global variational theory in fibred spaces, The results on the structure of variational first order source forms can be applied to energy-momentum tensors, known in the general relativity, and field theory. The Euler equation (1.5.22) implies that the external potential v(r) in open molecular systems is determined by ρ only to a constant μ, which can be related to the chemical potential of an external electron reservoir (ℜ): μ = μℜ.

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