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Evaluation of basic integrals over Gaussian functions, traditionally utilized for electronic structure computations on molecules and solids, is discussed in a pedagogical form.
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By using Poissons summation formula, we calculate periodic integrals over Gaussian basis functions by partitioning the lattice summations between the real and reciprocal space, where both sums converge exponentially fast with a large exponent. We demonstrate that the summation can be performed efficiently to calculate 2-center Gaussian integrals over various kernels including overlap, kinetic, and Coulomb. The summation in real space is performed using an efficient flavor of the McMurchie-Davidson Recurrence Relation (MDRR). The expressions for performing summation in the reciprocal space are also derived and implemented. The algorithm for reciprocal space summation allows us to reuse several terms and leads to significant improvement in efficiency when highly contracted basis functions with large exponents are used. We find that the resulting algorithm is only between a factor of 5 to 15 slower than that for molecular integrals, indicating the very small number of terms needed in both the real and reciprocal space summations. An outline of the algorithm for calculating 3-center Coulomb integrals is also provided.
A measurement of the magnitude of the electric dipole moment of the electron (eEDM) larger than that predicted by the Standard Model (SM) of particle physics is expected to have a huge impact on the search for physics beyond the SM. Polar diatomic molecules containing heavy elements experience enhanced sensitivity to parity ($P$) and time-reversal ($T$)-violating phenomena, such as the eEDM and the scalar-pseudoscalar (S-PS) interaction between the nucleons and the electrons, and are thus promising candidates for measurements. The NL-textit{e}EDM collaboration is preparing an experiment to measure the eEDM and S-PS interaction in a slow beam of cold BaF molecules [Eur. Phys. J. D, 72, 197 (2018)]. Accurate knowledge of the electronic structure parameters, $W_d$ and $W_s$, connecting the eEDM and the S-PS interaction to the measurable energy shifts is crucial for the interpretation of these measurements. In this work we use the finite field relativistic coupled cluster approach to calculate the $W_d$ and $W_s$ parameters in the ground state of the BaF molecule. Special attention was paid to providing a reliable theoretical uncertainty estimate based on investigations of the basis set, electron correlation, relativistic effects and geometry. Our recommended values of the two parameters, including conservative uncertainty estimates, are 3.13 $pm$ $0.12 times 10^{24}frac{text{Hz}}{ecdot text{cm}}$ for $W_d$ and 8.29 $pm$ 0.12 kHz for $W_s$.
We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable molecules may be viable in the near future.
In this work, we present a linear optical implementation for analog quantum simulation of molecular vibronic spectra, incorporating the non-Condon scattering operation with a quadratically small truncation error. Thus far, analog and digital quantum algorithms for achieving quantum speedup have been suggested only in the Condon regime, which refers to a transition dipole moment that is independent of nuclear coordinates. For analog quantum optical simulation beyond the Condon regime (i.e., non-Condon transitions) the resulting non-unitary scattering operations must be handled appropriately in a linear optical network. In this paper, we consider the first and second-order Herzberg-Teller expansions of the transition dipole moment operator for the non-Condon effect, for implementation on linear optical quantum hardware. We believe the method opens a new way to approximate arbitrary non-unitary operations in analog and digital quantum simulations. We report in-silico simulations of the vibronic spectra for naphthalene, phenanthrene, and benzene to support our findings.
We present a quantum algorithm for calculating the vibronic spectrum of a molecule, a useful but classically hard problem in chemistry. We show several advantages over previous quantum approaches: vibrational anharmonicity is naturally included; after measurement, some state information is preserved for further analysis; and there are potential error-related benefits. Considering four triatomic molecules, we numerically study truncation errors in the harmonic approximation. Further, in order to highlight the fact that our quantum algorithms primary advantage over classical algorithms is in simulating anharmonic spectra, we consider the anharmonic vibronic spectrum of sulfur dioxide. In the future, our approach could aid in the design of materials with specific light-harvesting and energy transfer properties, and the general strategy is applicable to other spectral calculations in chemistry and condensed matter physics.
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