# Complete basis set extrapolation of electronic correlation energies using the Riemann zeta function

## TL;DR

This paper introduces a novel CBS extrapolation method for correlation energies using the Riemann zeta function, providing more accurate results without empirical parameters by analytically summing higher angular momentum contributions.

## Contribution

The paper presents an analytic CBS extrapolation technique based on the Riemann zeta function, improving accuracy over traditional methods without requiring empirical fitting.

## Key findings

- Zeta-function extrapolation yields smaller errors than conventional schemes.
- The method is easy to implement and parameter-free.
- Residual errors can be estimated reliably.

## Abstract

In this communication we present a method of complete basis set (CBS) extrapolation of correlation energies obtained with a systematic sequence of one-electron basis sets. Instead of fitting the finite-basis results with a certain functional form, we perform analytic re-summation of the missing contributions coming from higher angular momenta, $l$. The assumption that they vanish asymptotically as an inverse power of $l$ leads to an expression for the CBS limit given in terms of the Riemann zeta function. This result is turned into an extrapolation method that is very easy to use and requires no "empirical" parameters to be optimized. The performance of the proposed method is assessed by comparing the results with accurate reference data obtained with explicitly correlated theories and with results obtained with standard extrapolation schemes. On average, the errors of the zeta-function extrapolation are several times smaller compared with the conventional schemes employing the same number of points. A recipe for estimation of the residual extrapolation error is also proposed.