We present a strategy to explore the free energy landscapes of chemical reactions with post-transition-state bifurcations using an enhanced sampling method based on well-tempered metadynamics. Obviating the need for computationally expensive density functional theory-level ab initio molecular dynamics simulations, we obtain accurate energetics by utilizing a free energy perturbation scheme and deep learning estimator for the single-point energies of substrate configurations. Using a pair of easily interpretable collective variables, we present a quantitative free energy surface that is compatible with harmonic transition state theory calculations and in which the bifurcations are clearly visible. We demonstrate our approach with the example of the SpnF-catalyzed Diels–Alder reaction, a cycloaddition reaction in which post-transition-state bifurcation leads to the [4+2] as well as the [6+4] cycloadduct. We obtain the free energy landscapes for different stereochemical reaction pathways and characterize the mechanistic continuum between relevant reaction channels without explicitly searching for the pertinent transition state structures.
One of the hurdles in teaching undergraduate thermodynamics is a plethora of complicated partial derivative identities. Students suffer from difficulties in deriving, justifying, or interpreting the identities, misconceptions about partial derivatives, and a lack of in-depth understanding of the meanings of identities. We propose a novel diagrammatic method for the calculus of differentials and partial derivatives called the ‘sunray diagram’ that resolves the difficulties above. The sunray diagram technique relates a partial derivative with ‘arrow sliding’, which enables an aesthetic and intuitive manipulation of partial derivative expressions in the form of successive arrow slidings. Furthermore, the sunray diagram is more than an ad hoc or abstract machinery but is based on the symplectic structure of thermodynamics; the sunray diagram admits a direct physical interpretation on the P–V (or T–S) plane. We elaborate on such physical semantics of the sunray diagram by taking Maxwell’s approach to the geometry of thermodynamic structures—reinterpreted in terms of differential geometry—as a reference point. We anticipate that our discussion introduces the geometry of thermodynamics to learners and enriches the graphical pedagogy in physics education.
One of the hurdles in learning thermodynamics is a plethora of complicated partial derivative identities. Students suffer from difficulties in deriving, justifying, memorizing, or interpreting the identities, misconceptions about partial derivatives, and a lack of deeper understandings about the meaning of the identities. Here, we propose a diagrammatic method, the "sunray diagram," for the calculus of differentials and partial derivatives that resolves all of the aforementioned difficulties. With the sunray diagram, partial derivative identities can be instantly obtained in an intuitive manner by sliding arrows. Furthermore, the sunray diagram is more than an ad hoc machinery but based on the geometric structure of thermodynamics and admits direct physical interpretation on the P-V (or T-S) plane. Employing the language of differential forms and symplectic geometry, we show that the sunray diagram and Maxwell's previous work utilizing equal-area sliding of parallelograms are different visualizations of the same mathematical syntax, while the sunray diagram being more convenient in practice. We anticipate that our discussion introduces the geometry of thermodynamics to learners and enriches the graphical pedagogy in physics education.
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