Decoding heat capacity features from the energy landscape

A general scheme is derived to connect transitions in configuration space with features in the heat capacity. A formulation in terms of occupation probabilities for local minima that define the potential energy landscape provides a quantitative description of how contributions arise from competition between different states. The theory does not rely on a structural interpretation for the local minima, so it is equally applicable to molecular energy landscapes and the landscapes defined by abstract functions. Applications are presented for low-temperature solid-solid transitions in atomic clusters, which involve just a few local minima with different morphologies, and for cluster melting, which is driven by the landscape entropy associated with the more numerous high-energy minima. Analyzing these features in terms of the balance between states with increasing and decreasing occupation probabilities provides a direct interpretation of the underlying transitions. This approach enables us to identify a qualitatively different transition that is caused by a single local minimum associated with an exceptionally large catchment volume in configuration space for a machine learning landscape.

Figure 1

(a) $C_V$ for ${\mathrm{LJ}}_{13}$ calculated using parallel tempering (PT) and the harmonic superposition approximation (HSA) [6, 11, 16, 17]. (b) $C_V$ for the machine learning landscape obtained by fitting a three-layer neural network to predict patient outcomes from a combination of vital signs. (c) $C_V$ for ${\mathrm{LJ}}_{31}$ calculated using basin sampling [24]. The lowest and second-lowest minima with Mackay and anti-Mackay overlayers [27] colored in green are illustrated on the left and right of the low-temperature peak, respectively. The 13-atom icosahedral core is colored red in both structures. (d) $C_V$ for ${\mathrm{LJ}}_{75}$ calculated using basin sampling [24]. The lowest and second-lowest minima based on a Marks decahedron and an incomplete Mackay icosahedron are illustrated on the left and right of the low-temperature peak, respectively. The atoms are colored according to their contribution to the total energy: The most tightly bound atoms are blue, the least tightly bound are red, with intermediate binding energies in green.

Figure 2

Disconnectivity graphs for ${\mathrm{LJ}}_{13}$ colored according to the occupation probability gradients projected onto local minima, as derived in Eq. (18). Minima are colored for contributions to the heat capacity that account for (a) 90% and (b) 99% of the contributions from positive (blue) and negative (red) gradients at $k_{B}T/\epsilon =0.28$. The two-dimensional probability distributions required were obtained by basin sampling [24]. For comparison, the results for ${\mathrm{LJ}}_{14}$ in (c) and (d) correspond to assignments based on harmonic superposition analysis for the occupation probabilities of local minima [Eq. (12)] with the same thresholds of (c) 90% and (d) 99%. The global minima are illustrated by the corresponding branches in each case. The corresponding stationary point databases include 1441 minima and 21 161 transition states for ${\mathrm{LJ}}_{13}$, with 3413 minima and 52 005 transition states for ${\mathrm{LJ}}_{14}$.

Figure 3

Disconnectivity graphs for double-funnel landscapes colored according to the overall occupation probability gradients for the quench energy bin associated with each local minimum, as derived in Eq. (18). Minima are colored for contributions to the heat capacity that account for 99% of the contributions from positive (blue) and negative (red) gradients. The two-dimensional probability distributions required were obtained by basin sampling [24] and the corresponding stationary point databases include 47 193 minima and 100 704 transition states for ${\mathrm{LJ}}_{31}$, with 312 178 minima and 602 580 transition states for ${\mathrm{LJ}}_{75}$. (a) ${\mathrm{LJ}}_{31}$ at $k_{B}T/\epsilon =0.0268$, (b) ${\mathrm{LJ}}_{31}$ at $k_{B}T/\epsilon =0.329$, (c) ${\mathrm{LJ}}_{75}$ at $k_{B}T/\epsilon =0.082$, (d) ${\mathrm{LJ}}_{75}$ at $k_{B}T/\epsilon =0.291$. For the low-temperature peaks corresponding to solid-solid transitions the assignments are the same as for the harmonic superposition analysis applied directly to the occupation probabilities of local minima, as defined in Eq. (12). For the higher-temperature melting peaks, neglect of well anharmonicity in a harmonic analysis systematically shifts positive probability gradients to higher-energy minima (not shown). The lowest and second-lowest minima with Mackay and anti-Mackay overlayers [27] are illustrated for ${\mathrm{LJ}}_{31}$ in (a) and (b). For ${\mathrm{LJ}}_{75}$ the global minimum Marks decahedron and the second-lowest minimum based on icosahedral packing with an anti-Mackay overlayer are illustrated in (c). The graphs in (b) and (d) exclude all minima that do not contribute to components of the heat capacity peak within the 99% threshold, and the global minimum does not appear for ${\mathrm{LJ}}_{75}$ in (d). Instead, the lowest minimum based on icosahedral packing with a Mackay overlayer is illustrated along with the second-lowest minimum, both of which belong to the set with the negative occupation probability gradient.

Figure 4

Disconnectivity graphs for a machine learning landscape corresponding to the heat capacity analog illustrated in Fig. 1. (a) is the complete graph including all the minima. (b) and (c) correspond to the temperatures of the two peaks in $C_V$ at (b) $k_{B}T=0.000236$, (b) $k_{B}T=0.000761$, including only minima that account for 90% of the contributions from positive (blue) and negative (red) gradients based on harmonic superposition analysis for the occupation probabilities [Eq. (12)]. The vertical position of each minimum is the same in the three panels and corresponds to the value of the cost function. The stationary point database contains 1997 minima and 7492 transition states.