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papers/bpm_L.md

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+---
+author: Anagha Aneesh
+date: 2024-11-20
+bibliography: ../references.bib
+---
+
+# A Bond-Based Machine Learning Model for Molecular Polarizabilities and A Priori Raman Spectra
+
+## Why I chose this paper
+
+- Reconstructing IR/Raman spectra is interesting to me
+- Applications of ML to electronic structure theory (outside of potential) is cool
+
+## Introduction
+
+- Machine learning force fields (ML FF) have accelerated the field of molecular simulations – specifically for systems where there is not an established force field
+- ML FFs are significantly more cost-efficient
+- Learning molecular properties like dipole moment (IR) and polarizability (Raman) can help in interpreting spectra signals and benchmarking ML accuracy against experiment
+- Neural network algorithms v.s. kernel algorithms
+  - NN: better performance + lower cost for larger systems
+  - kernel: requires less data + high cost for large systems
+
+## Existing Work on ML Models for Electric Polarizability
+
+- Equivariant neural networks, response formalism, Applequists's dipole interaction model
+- Two kernel-based methods
+  - align structures in training data and treat tensor components as scalars
+    - requires lots of data
+  - symmetry-adapted Gaussian regression
+    - generalization of scalar kernel ridge regression (KRR)
+
+### Goal of this work
+
+- KRR on bond polarizability model (BPM)
+  - molecular polarizability is a sum over bond contributions
+
+## Theory
+
+### Bond Polarizability Model
+
+- total molecular polarizability, $\alpha$ is the sum of bond polarizabilities
+
+$$
+\alpha = \sum_b{\alpha^b}
+$$
+
+- elements of individual bond polarizability tensors
+
+$$
+\alpha_{ij}^{b} = \frac{1}{3}(2\alpha_p^b + \alpha_l^b)\delta_{ij} + (\alpha_l^b + \alpha_p^b)(\hat{R}_i^b\hat{R}_j^b - \frac{1}{3}\delta_{ij})
+$$
+
+- assumes bonds are cylindrically symmetric and typically assumes total polarizability only depends on bond length
+
+## ML Model
+
+- Separate polarizability tensor into isotropic and anisotropic components so the ML task is to infer these
+
+$$
+\alpha = \alpha_{\text{iso}}\bf{1}+\beta
+$$
+
+- Rewrite elements of tensor in terms of components
+
+$$
+\alpha_{ij}^{b} = \alpha_{\text{iso}}^b\delta_{ij} + \beta^b(\hat{R}_i^b\hat{R}_j^b - \frac{1}{3}\delta_{ij})
+$$
+
+- KRR used to evaluate isotropic component, summed over bonds instead of atoms
+
+$$
+\alpha_{\text{iso}} = \sum_b{\alpha_{\text{iso}}^b} = \sum_n{\sum{K(\textbf{q}^b},\textbf{q}^{b'})w_n}
+$$
+
+- Using a Gaussian kernel
+
+$$
+K(\textbf{q}^b,\textbf{q}^{b'}) = \text{exp}(-\gamma||\textbf{q}^b-\textbf{q}^{b'}||^2)
+$$
+
+- The same can be done for anisotropic component
+
+$$
+\beta_{ij} = \sum_b{\beta^bQ_{ij}^b} = \sum_n{\sum_{b,b'}{K(\textbf{q}^b},\textbf{q}^{b'})Q_{ij}^bv_n}
+$$
+
+- loss function
+
+$$
+\mathcal{L} = \frac{1}{2}\sum_{i,j}{||\beta_{ij} - \textbf{K}_{ij}\textbf{v}||^2}
+$$
+
+## Raman Spectra
+
+- Calculating the anharmonic IR and Raman spectra
+
+$$
+I_{\text{iso}}(\omega) \propto v(\omega) \int{dt \ e^{i\omega t}\langle\dot{\alpha}_{\text{iso}}(\tau)\dot{\alpha}_{\text{iso}}(t-\tau)}\rangle_\tau
+$$
+$$
+I_{\text{aniso}}(\omega) \propto v(\omega) \int{dt \ e^{i\omega t}\langle Tr[\dot{\beta}_{\text{iso}}(\tau)\dot{\beta}_{\text{iso}}(t-\tau)}]\rangle_\tau
+$$
+
+## Biphenyl
+
+![biphenyl](./bpm_ml_figures/bpm_fig1.webp)
+
+## Raman spectra evaluation
+
+![spectra](./bpm_ml_figures/bpm_fig2.webp)
+
+## Malonaldehyde
+
+- keto and enol forms
+
+![malonaldehyde](./bpm_ml_figures/bpm_fig3.webp)
+
+## Future Directions
+
+- Deep neural network implementation of BPM
+- Consider all bonds within a cutoff region