Computing Second-Order Functional Derivatives with Respect to the External Potential
Bibliographic record
Abstract
Following the increasing interest in the higher-order (functional) derivatives of conceptual density functional theory, we developed and implemented a method for calculating second-order functional derivatives with respect to the external potential. Our method is theoretically exact but involves two numerical approximations: the functional derivatives are expanded in a basis set, and the values of the corresponding expansion coefficients are determined by probing the molecular environment by a finite set of external potential perturbations. Exact solutions are obtained only in the limit of a complete basis set and an infinite number of distinct perturbations. We use this method to compute the atom-condensed linear response kernel for a series of six molecules and show that the results are comparable to the ones obtained by a previously proposed, approximate approach from second-order perturbation theory. The numerical error of the current implementation is about 0.01 au. Because the present method gives exact or quasi-exact solutions, it can be used as a benchmark against which approximate approaches are assessed.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".