Thermodynamic integration - Revision history

Csheldon: /* Related tags and articles */

2026-06-19T10:20:25Z

Csheldon: /* Thermodynamic integration using internal coordinates (TILAMBDA) */

2026-06-15T08:01:42Z

Thermodynamic integration using internal coordinates (TILAMBDA)

← Older revision		Revision as of 08:01, 15 June 2026
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	Unfortunately, there are several problems linked with such a straightforward approach. First, the systems with rotational and/or translational degrees of freedom cannot be treated in a straightforward manner because <math>V_{0,\mathbf{x}}(\mathbf{x})</math> is not invariant under rotations and translations. Conventional TI is thus unsuitable for simulations of gas-phase molecules or adsorbate-substrate systems. and this problem also imposes restrictions on the choice of thermostat used in NVT simulation (Langevin thermostat, for instance, does not conserve the position of the center of mass and is therefore unsuitable for use in conventional TI). Furthermore, if the Hesse matrix of the harmonic system has one or more eigenvalues that nearly vanish, the simulations with <math>\lambda \rightarrow </math> 0 are likely to generate unphysical configurations, causing serious convergence issues.		Unfortunately, there are several problems linked with such a straightforward approach. First, the systems with rotational and/or translational degrees of freedom cannot be treated in a straightforward manner because <math>V_{0,\mathbf{x}}(\mathbf{x})</math> is not invariant under rotations and translations. Conventional TI is thus unsuitable for simulations of gas-phase molecules or adsorbate-substrate systems. and this problem also imposes restrictions on the choice of thermostat used in NVT simulation (Langevin thermostat, for instance, does not conserve the position of the center of mass and is therefore unsuitable for use in conventional TI). Furthermore, if the Hesse matrix of the harmonic system has one or more eigenvalues that nearly vanish, the simulations with <math>\lambda \rightarrow </math> 0 are likely to generate unphysical configurations, causing serious convergence issues.

	== Thermodynamic integration using internal coordinates ~~(TILAMBDA)~~ ==		== Thermodynamic integration using internal coordinates ==
	The problems of TI using a harmonic reference have been addressed in a series of works by Amsler et al {{Cite\|bucko:studt:jctc:2021}}{{Cite\|bucko:studt:jctc:2023}}. First, the method was formulated in terms of rotationally and translationally invariant internal coordinates <math>\mathbf{q}=\mathbf{q}(\mathbf{x})</math>, whereby the free energy of the interacting system is repartitioned as follows:		The problems of TI using a harmonic reference have been addressed in a series of works by Amsler et al {{Cite\|bucko:studt:jctc:2021}}{{Cite\|bucko:studt:jctc:2023}}. First, the method was formulated in terms of rotationally and translationally invariant internal coordinates <math>\mathbf{q}=\mathbf{q}(\mathbf{x})</math>, whereby the free energy of the interacting system is repartitioned as follows:
	:<math>		:<math>

Csheldon: /* Thermodynamic integration with harmonic reference (SCALEE) */

2026-06-15T08:01:31Z

Thermodynamic integration with harmonic reference (SCALEE)

← Older revision		Revision as of 08:01, 15 June 2026
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	There are many options that can be taken for the non-interacting and interacting systems. In Dorner et al., TI was first performed from a harmonic system (the ideal gas) to the full ''ab initio'' Hamiltonian (e.g., PBE) to include the anharmonic contributions {{Cite\|dorner:PRL:2018}}; a second TI was then performed to increase the ''k''-mesh density and change the density functional (e.g., SCAN, HSE06). We will now describe the theory when going from a harmonic to anharmonic system, as this requires some special details.		There are many options that can be taken for the non-interacting and interacting systems. In Dorner et al., TI was first performed from a harmonic system (the ideal gas) to the full ''ab initio'' Hamiltonian (e.g., PBE) to include the anharmonic contributions {{Cite\|dorner:PRL:2018}}; a second TI was then performed to increase the ''k''-mesh density and change the density functional (e.g., SCAN, HSE06). We will now describe the theory when going from a harmonic to anharmonic system, as this requires some special details.

	== Thermodynamic integration with harmonic reference ~~(SCALEE)~~ ==		== Thermodynamic integration with harmonic reference ==
	The Helmholtz free energy (<math>A</math>) of a fully interacting system (1) can be expressed in terms of that of system harmonic in Cartesian coordinates (0,<math>\mathbf{x}</math>) as follows		The Helmholtz free energy (<math>A</math>) of a fully interacting system (1) can be expressed in terms of that of system harmonic in Cartesian coordinates (0,<math>\mathbf{x}</math>) as follows
	:<math>		:<math>

Csheldon: /* Thermodynamic integration with harmonic reference */

2026-06-09T08:45:04Z

Thermodynamic integration with harmonic reference

← Older revision		Revision as of 08:45, 9 June 2026
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	There are many options that can be taken for the non-interacting and interacting systems. In Dorner et al., TI was first performed from a harmonic system (the ideal gas) to the full ''ab initio'' Hamiltonian (e.g., PBE) to include the anharmonic contributions {{Cite\|dorner:PRL:2018}}; a second TI was then performed to increase the ''k''-mesh density and change the density functional (e.g., SCAN, HSE06). We will now describe the theory when going from a harmonic to anharmonic system, as this requires some special details.		There are many options that can be taken for the non-interacting and interacting systems. In Dorner et al., TI was first performed from a harmonic system (the ideal gas) to the full ''ab initio'' Hamiltonian (e.g., PBE) to include the anharmonic contributions {{Cite\|dorner:PRL:2018}}; a second TI was then performed to increase the ''k''-mesh density and change the density functional (e.g., SCAN, HSE06). We will now describe the theory when going from a harmonic to anharmonic system, as this requires some special details.

	== Thermodynamic integration with harmonic reference ==		== Thermodynamic integration with harmonic reference (SCALEE) ==
	The Helmholtz free energy (<math>A</math>) of a fully interacting system (1) can be expressed in terms of that of system harmonic in Cartesian coordinates (0,<math>\mathbf{x}</math>) as follows		The Helmholtz free energy (<math>A</math>) of a fully interacting system (1) can be expressed in terms of that of system harmonic in Cartesian coordinates (0,<math>\mathbf{x}</math>) as follows
	:<math>		:<math>
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	# integrate <math>\langle V_1 -V_{0,\mathbf{x}} \rangle</math> over the <math>\lambda </math> grid and compute <math>\Delta A_{0,\mathbf{x}\rightarrow 1}</math>		# integrate <math>\langle V_1 -V_{0,\mathbf{x}} \rangle</math> over the <math>\lambda </math> grid and compute <math>\Delta A_{0,\mathbf{x}\rightarrow 1}</math>

	Unfortunately, there are several problems linked with such a straightforward approach. First, the systems with rotational and/or translational degrees of freedom cannot be treated in a straightforward manner because <math>V_{0,\mathbf{x}}(\mathbf{x})</math> is not invariant under rotations and translations. Conventional TI is thus unsuitable for simulations of gas-phase molecules or adsorbate-substrate systems. and this problem also imposes restrictions on the choice of thermostat used in NVT simulation (Langevin thermostat, for instance, does not conserve the position of the center of mass and is therefore unsuitable for use in conventional TI). Furthermore, if the Hesse matrix of the harmonic system has one or more eigenvalues that nearly vanish, the simulations with <math>\lambda \rightarrow </math> 0 are likely to generate unphysical configurations, causing serious convergence issues.		Unfortunately, there are several problems linked with such a straightforward approach. First, the systems with rotational and/or translational degrees of freedom cannot be treated in a straightforward manner because <math>V_{0,\mathbf{x}}(\mathbf{x})</math> is not invariant under rotations and translations. Conventional TI is thus unsuitable for simulations of gas-phase molecules or adsorbate-substrate systems. and this problem also imposes restrictions on the choice of thermostat used in NVT simulation (Langevin thermostat, for instance, does not conserve the position of the center of mass and is therefore unsuitable for use in conventional TI). Furthermore, if the Hesse matrix of the harmonic system has one or more eigenvalues that nearly vanish, the simulations with <math>\lambda \rightarrow </math> 0 are likely to generate unphysical configurations, causing serious convergence issues.

	== Thermodynamic integration using internal coordinates (TILAMBDA) ==		== Thermodynamic integration using internal coordinates (TILAMBDA) ==

Csheldon: /* Thermodynamic integration using a Hessian reference */

2026-06-09T08:44:54Z

Thermodynamic integration using a Hessian reference

← Older revision		Revision as of 08:44, 9 June 2026
Line 52:		Line 52:
	Unfortunately, there are several problems linked with such a straightforward approach. First, the systems with rotational and/or translational degrees of freedom cannot be treated in a straightforward manner because <math>V_{0,\mathbf{x}}(\mathbf{x})</math> is not invariant under rotations and translations. Conventional TI is thus unsuitable for simulations of gas-phase molecules or adsorbate-substrate systems. and this problem also imposes restrictions on the choice of thermostat used in NVT simulation (Langevin thermostat, for instance, does not conserve the position of the center of mass and is therefore unsuitable for use in conventional TI). Furthermore, if the Hesse matrix of the harmonic system has one or more eigenvalues that nearly vanish, the simulations with <math>\lambda \rightarrow </math> 0 are likely to generate unphysical configurations, causing serious convergence issues.		Unfortunately, there are several problems linked with such a straightforward approach. First, the systems with rotational and/or translational degrees of freedom cannot be treated in a straightforward manner because <math>V_{0,\mathbf{x}}(\mathbf{x})</math> is not invariant under rotations and translations. Conventional TI is thus unsuitable for simulations of gas-phase molecules or adsorbate-substrate systems. and this problem also imposes restrictions on the choice of thermostat used in NVT simulation (Langevin thermostat, for instance, does not conserve the position of the center of mass and is therefore unsuitable for use in conventional TI). Furthermore, if the Hesse matrix of the harmonic system has one or more eigenvalues that nearly vanish, the simulations with <math>\lambda \rightarrow </math> 0 are likely to generate unphysical configurations, causing serious convergence issues.

	== Thermodynamic integration using ~~a Hessian reference~~ ==		== Thermodynamic integration using internal coordinates (TILAMBDA) ==
	The problems of TI using a harmonic reference have been addressed in a series of works by Amsler et al {{Cite\|bucko:studt:jctc:2021}}{{Cite\|bucko:studt:jctc:2023}}. First, the method was formulated in terms of rotationally and translationally invariant internal coordinates <math>\mathbf{q}=\mathbf{q}(\mathbf{x})</math>, whereby the free energy of the interacting system is repartitioned as follows:		The problems of TI using a harmonic reference have been addressed in a series of works by Amsler et al {{Cite\|bucko:studt:jctc:2021}}{{Cite\|bucko:studt:jctc:2023}}. First, the method was formulated in terms of rotationally and translationally invariant internal coordinates <math>\mathbf{q}=\mathbf{q}(\mathbf{x})</math>, whereby the free energy of the interacting system is repartitioned as follows:
	:<math>		:<math>

Csheldon: /* Basic theory */

2026-06-09T08:44:20Z

Basic theory

← Older revision		Revision as of 08:44, 9 June 2026
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	The free energy of a fully interacting system can be written as the sum of the free energy a non-interacting reference system and the difference in the free energy of the fully interacting system and the non-interacting system {{Cite\|dorner:PRL:2018}}		The free energy of a fully interacting system can be written as the sum of the free energy a non-interacting reference system and the difference in the free energy of the fully interacting system and the non-interacting system {{Cite\|dorner:PRL:2018}}

	::<math> F_{1} = F_{0} + \Delta F_{0\rightarrow 1} </math>.		::<math> A_{1} = A_{0} + \Delta A_{0\rightarrow 1} </math>.

	Using thermodynamic integration the free energy difference between the two systems is written as		Using thermodynamic integration the free energy difference between the two systems is written as

	<math> \Delta F_{0\rightarrow 1} = \int\limits_{0}^{1} d\lambda \langle U_{1}(\lambda) - U_{0}(\lambda) \rangle_{\lambda} </math>.		<math> \Delta A_{0\rightarrow 1} = \int\limits_{0}^{1} d\lambda \langle U_{1}(\lambda) - U_{0}(\lambda) \rangle_{\lambda} </math>.

	Here <math>U_{1}(\lambda)</math> and <math>U_{0}(\lambda)</math> describe the potential energies of a fully-interacting and a non-interacting reference system, respectively. The coupling strength of the systems is controlled via the coupling parameter <math>\lambda</math>. It is neccessary that the connection of the two systems via the coupling constant is reversible. The notation <math>\langle \ldots \rangle_{\lambda}</math> denotes an ensemble average of a system driven by the following classical Hamiltonian		Here <math>U_{1}(\lambda)</math> and <math>U_{0}(\lambda)</math> describe the potential energies of a fully-interacting and a non-interacting reference system, respectively. The coupling strength of the systems is controlled via the coupling parameter <math>\lambda</math>. It is neccessary that the connection of the two systems via the coupling constant is reversible. The notation <math>\langle \ldots \rangle_{\lambda}</math> denotes an ensemble average of a system driven by the following classical Hamiltonian

Csheldon at 14:11, 28 May 2026

2026-05-28T14:11:13Z

← Older revision		Revision as of 14:11, 28 May 2026
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	== Related tags and articles ==		== Related tags and articles ==
	{{TAG\|VCAIMAGES}}, {{TAG\|NCORE_IN_IMAGE1}}, {{TAG\|IMAGE_N}}, {{TAG\|SCALEE}}, {{FILE\|DYNMATFULL}}, {{TAG\|PHON_NSTRUCT}}, {{TAG\|TILAMBDA}}, {{FILE\|HESSEMAT}}, {{FILE\|ICONST}}, {{FILE\|REPORT}}		{{TAG\|VCAIMAGES}}, {{TAG\|NCORE_IN_IMAGE1}}, {{TAG\|IMAGE_N}}, {{TAG\|SCALEE}}, {{FILE\|DYNMATFULL}}, {{TAG\|PHON_NSTRUCT}}, {{TAG\|TILAMBDA}}, {{FILE\|HESSEMAT}}, {{FILE\|ICONST}}, {{FILE\|REPORT}}

			[[Thermodynamic integration calculations]]

	== References ==		== References ==

Csheldon: /* Related tags and articles */

2026-05-28T14:04:05Z

Csheldon at 14:03, 28 May 2026

2026-05-28T14:03:04Z

← Older revision		Revision as of 14:03, 28 May 2026
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	</math>		</math>
	being the Wilson matrix. Note that the calculation of the term <math>\Delta A_{0,\mathbf{x} \rightarrow 0,\mathbf{q}}</math> is inexpensive as it corresponds to a force field to force field transformation. Furthermore, this term vanishes in the case of phase volume conserving coordinates, such as interatomic distances.		being the Wilson matrix. Note that the calculation of the term <math>\Delta A_{0,\mathbf{x} \rightarrow 0,\mathbf{q}}</math> is inexpensive as it corresponds to a force field to force field transformation. Furthermore, this term vanishes in the case of phase volume conserving coordinates, such as interatomic distances.

			== Related tags and articles ==
			{{TAG\|VCAIMAGES}}, {{TAG\|NCORE_IN_IMAGE1}}, {{TAG\|IMAGE_N}}, {{TAG\|SCALEE}}, {{FILE\|DYNMATFULL}}, {{TAG\|TILAMBDA}}, {{FILE\|HESSEMAT}}, {{FILE\|ICONST}}, {{FILE\|REPORT}}

	== References ==		== References ==

Csheldon: /* Thermodynamic integration using a Hessian reference */

2026-03-25T14:57:12Z

Thermodynamic integration using a Hessian reference

← Older revision		Revision as of 14:57, 25 March 2026
Line 53:		Line 53:

	== Thermodynamic integration using a Hessian reference ==		== Thermodynamic integration using a Hessian reference ==
	~~These~~ problems of TI using a harmonic reference have been addressed in a series of works by Amsler et al {{Cite\|bucko:studt:jctc:2021}}{{Cite\|bucko:studt:jctc:2023}}. First, the method was formulated in terms of rotationally and translationally invariant internal coordinates <math>\mathbf{q}=\mathbf{q}(\mathbf{x})</math>, whereby the free energy of the interacting system is repartitioned as follows:		The problems of TI using a harmonic reference have been addressed in a series of works by Amsler et al {{Cite\|bucko:studt:jctc:2021}}{{Cite\|bucko:studt:jctc:2023}}. First, the method was formulated in terms of rotationally and translationally invariant internal coordinates <math>\mathbf{q}=\mathbf{q}(\mathbf{x})</math>, whereby the free energy of the interacting system is repartitioned as follows:
	:<math>		:<math>
	A_1 = A_{0,\mathbf{x}} + \Delta A_{0,\mathbf{x} \rightarrow 0,\mathbf{q}} + \Delta A_{0,\mathbf{q} \rightarrow 1}		A_1 = A_{0,\mathbf{x}} + \Delta A_{0,\mathbf{x} \rightarrow 0,\mathbf{q}} + \Delta A_{0,\mathbf{q} \rightarrow 1}
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	\mathbf{\underline{B}}_{i,j} = \frac{\partial q_i}{\partial x_j}		\mathbf{\underline{B}}_{i,j} = \frac{\partial q_i}{\partial x_j}
	</math>		</math>
	being the Wilson matrix. Note that the calculation of the term <math>\Delta A_{0,\mathbf{x} \rightarrow 0,\mathbf{q}}</math> is inexpensive as it corresponds to a force field to force field transformation. Furthermore, this term vanishes in the case of phase volume conserving coordinates, such as interatomic distances.		being the Wilson matrix. Note that the calculation of the term <math>\Delta A_{0,\mathbf{x} \rightarrow 0,\mathbf{q}}</math> is inexpensive as it corresponds to a force field to force field transformation. Furthermore, this term vanishes in the case of phase volume conserving coordinates, such as interatomic distances.

	== References ==		== References ==

← Older revision		Revision as of 10:20, 19 June 2026
Line 72:		Line 72:

	== Related tags and articles ==		== Related tags and articles ==
	{{TAG\|VCAIMAGES}}, {{TAG\|NCORE_IN_IMAGE1}}, {{TAG\|~~IMAGE_N~~}}, {{TAG\|SCALEE}}, {{FILE\|DYNMATFULL}}, {{TAG\|PHON_NSTRUCT}}, {{TAG\|TILAMBDA}}, {{FILE\|HESSEMAT}}, {{FILE\|ICONST}}, {{FILE\|REPORT}}		{{TAG\|VCAIMAGES}}, {{TAG\|NCORE_IN_IMAGE1}}, {{TAG\|IMAGE_1}}, {{TAG\|SCALEE}}, {{FILE\|DYNMATFULL}}, {{TAG\|PHON_NSTRUCT}}, {{TAG\|TILAMBDA}}, {{FILE\|HESSEMAT}}, {{FILE\|ICONST}}, {{FILE\|REPORT}}

	[[Thermodynamic integration calculations]]		[[Thermodynamic integration calculations]]

← Older revision		Revision as of 14:04, 28 May 2026
Line 72:		Line 72:

	== Related tags and articles ==		== Related tags and articles ==
	{{TAG\|VCAIMAGES}}, {{TAG\|NCORE_IN_IMAGE1}}, {{TAG\|IMAGE_N}}, {{TAG\|SCALEE}}, {{FILE\|DYNMATFULL}}, {{TAG\|TILAMBDA}}, {{FILE\|HESSEMAT}}, {{FILE\|ICONST}}, {{FILE\|REPORT}}		{{TAG\|VCAIMAGES}}, {{TAG\|NCORE_IN_IMAGE1}}, {{TAG\|IMAGE_N}}, {{TAG\|SCALEE}}, {{FILE\|DYNMATFULL}}, {{TAG\|PHON_NSTRUCT}}, {{TAG\|TILAMBDA}}, {{FILE\|HESSEMAT}}, {{FILE\|ICONST}}, {{FILE\|REPORT}}

	== References ==		== References ==