As early as 1926, Erwin Schrödinger postulated the time-dependent Schrödinger equation (1) which makes it possible to describe the temporal change of systems (atoms, molecules or even materials) completely non-relativistically. He thus formulated the foundations of quantum mechanics on which modern simulation technologies in computational chemistry are based. Within this theoretical construct, the atomic nuclei and electrons in the system under consideration are described as wave functions and interact with each other and the environment via the Hamilton operator.

Born-Oppenheimer approximation and Schrödinger equation

Considering the large mass difference between electrons and nuclei, a valid assumption can be made that electrons move much faster than nuclei. Within this Born-Oppenheimer approximation (2) it is possible to formulate a so-called time-independent electronic Schrödinger equation and to solve it parametrically for a given nuclear configuration (e.g. molecular structure). Thereby, one obtains solutions for discrete electronic states (electronic ground state and excited states) with stationary electronic wave functions and the corresponding electronic energies. This energy is also commonly referred to as the potential energy surface and is the energy landscape on which the nuclei of the molecular system move, for example during a chemical reaction. Although the introduction of these approximations has significantly reduced the complexity of the problem, the electronic problem still cannot be solved exactly for the vast majority of systems. Therefore, other approximations such as the mean-field method are used, where electrons are treated independently and interact with all other electrons via an effective potential. This simplifies the solution of the time-independent electronic Schrödinger equation from an N-electron problem to N one-electron problems. The total wave function can then be formulated in terms of a Slater determinant to guarantee its antisymmetry.

The concept of orbitals

In this approach, the concept of orbitals is introduced for the first time through the approximations. Orbitals are the wave functions of the individual electrons and the absolute square of an orbital describes their residence probability of an electron in the vicinity of an atomic nucleus. These orbitals are extended to molecular orbitals in the MO-LCAO approach (molecular orbitals as linear combinations of atomic orbitals) as part of the computer-aided solution of the Hartree-Fock-Roothaan-Hall equation (3, 4) and the equations are solved iteratively with the help of the variational principle. The efficient solution of these and similar quantum mechanical equations is at the heart of modern quantum chemistry programs.

Post-Hartree-Fock methods

Since the solution of the Hartree-Fock-Roothaan-Hall equation gives good results only within the approximations taken, several methods have been developed to improve the prediction quality. These Post-Hartree-Fock methods (Post-HF methods) achieve very accurate results compared to experimental values because the different interactions of the electrons (correlation energy) are described more accurately. The conceptually simplest of these methods is the so-called configuration interaction (CI) method. While this and other Post-HF methodologies (e.g., coupled-cluster method) allow accurate predictions of the ground state and excited states, the computational cost quickly becomes prohibitively large with increasing system size and is therefore only economical on classical hardware for small molecules. With the advent of quantum computers, it will potentially be possible in the future to solve larger systems using methods such as the quantum phase estimation (QPE) algorithm (5, 6) and the variational quantum eigensolver (VQE) (7) . Although the former provides the exact ground state with a controllable error, this approach requires a large number of error-tolerant qubits. For quantum computing in the near future - the so-called Noisy-Intermediate-Scale-Quantum (NISQ) era - the VQE approach seems more suitable since it is an iterative approach that optimizes short parameterized quantum circuits.

Density functional theory and related theorems.

Density functional theory (DFT) elegantly circumvents the explicit treatment of the correlated N-electron wavefunction by mapping the high-dimensional problem to the three-dimensional ground-state electron density. The use of the Hohenberg-Kohn theorems (8) provides the exact ground-state energy as a functional of this one-electron density and could be determined variationally, provided that all terms of the functional were known. However, since this is not the case, one usually resorts first to Kohn-Sham formulation of DFT (KS-DFT) (9) and formulates the one-electron density as a sum of one-particle densities whose solution is equivalent to that of the Hartree-Fock-Rothaan-Hall equation. Finally, the still unknown exchange-correlation potential can be approximated by different approaches. The simplest of these methods is the so-called local-density approximation (LDA) (10, 11), in which the exchange-correlation energy of the homogeneous electron gas is used. This approach neglects the rapidly changing electron density in molecular systems. An approximation that accounts for this phenomenon is the generalized gradient approximation (GGA), which includes an additional term for the derivative of the density. The results can be further improved by enriching the exchange-correlation energy proportionally with the exact exchange energy known from the Hartree-Fock approach (12). Although these hybrid functionals are much more computationally intensive than the GGA functionals, they are generally much less expensive than Post-HF methods and provide high-quality results. Following the Runge-Gross theorem (13) and the van Leeuwen theorem (14), the previous consideration of time-independent systems can be extended to time-dependent systems to form the Time-Dependent Density Functional Theory (TDDFT), and thus properties such as excited electronic states and associated optical properties such as UV/Vis spectra can also be determined.

Both GGA functionals, such as PBE (15), and hybrid functionals, such as B3LYP (16, 17) or PBE0 (18), have become standard in computational chemistry for the characterization of molecular systems due to their reliability and have also been routinely used for the simulation of realistic systems and problems since the early 2000s due to increased computational power.

Using the methods described, a wide range of chemical and physical properties can be reliably simulated. Combining such predictions with each other and furthermore applying methodologies to link simulations and their results in a meaningful way, even highly complex problems can be solved.

This is the central objective of our cloud-based SaaS platform. Not only does it take over the currently highly manual task of setting up chemical simulations, providing computing resources, running and evaluating them, it is an end-to-end solution in which the individual simulations are assembled like building blocks and intuitively merge into complex workflows.

With QuantistryLab, all you need to run chemical simulations is a web browser. Our cloud-native platform redefines R&D with a holistic computational approach, from quantum to AI. Our customized Use-Case Modules offer tailored solutions to overcome your specific R&D challenges. Get in touch and start simulating today.

Literature

(1) E. Schrödinger, "Quantization as an eigenvalue problem," Ann. Phys. (Leipzig) 81, 109 (1926).

(2) D. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective (University Science Books, Sausalito, Calif, 2007).

(3) C. C. J. Roothaan, "New developments in molecular orbital theory," Rev. Mod. Phys. 23, 69 (1951).

(4) G. G. Hall, "The molecular orbital theory of chemical valency. VIII. a method of calculating ionization potentials," Proc. R. Soc. Lond. A 205, 541 (1951).

(5) A. W. Harrow, A. Hassidim, S. Lloyd, "Quantum algorithm for linear systems of equations," Phys. Rev. Lett. 103, 150502 (2009).

(6) A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, D.A. Spielman, "Exponential algorithmic speedup by a quantum walk." (ACM Press, 2003).

(7) J. R. McClean, J. Romero, R. Babbush, A. Aspuru-Guzik, "The theory of variational hybrid quantum-classical algorithms," New J. Phys., 18, 023023 (2016).

(8) P. Hohenberg and W. Kohn, "Inhomogeneous electron gas," Phys. Rev. 136, B864 (1964).

(9) W. Kohn and L. J. Sham, "Self-consistent equations including exchange and correlation effects," Phys. Rev. 140, A1133 (1965).

(10) J. C. Slater, "A simplification of the Hartree-Fock method," Phys. Rev. 81, 385 (1951).

(11) J. P. Perdew and Y. Wang, "Accurate and simple analytic representation of the electron-gas correlation energy," Phys. Rev. B 45, 13244 (1992).

(12) A. D. Becke, "A new mixing of Hartree-Fock and local density-functional theories," J. Chem. Phys. 98, 1372 (1993).

(13) E. Runge and E. K. U. Gross, "Density-functional theory for time-dependent systems," Phys. Rev. Lett. 52, 997 (1984).

(14) R. van Leeuwen, "Mapping from densities to potentials in time-dependent density-functional theory," Phys. Rev. Lett. 82, 3863 (1999).

(15) J. P. Perdew, K. Burke, and M. Ernzerhof, "Generalized gradient approximation made simple," Phys. Rev. Lett. 77, 3865 (1996).

(16) A. D. Becke, "Density-functional thermochemistry. III. the role of exact exchange," J. Chem. Phys. 98, 5648 (1993).

(17) P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch, "Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields," J. Phys. Chem. 98, 11623 (1994).

(18) C. Adamo and V. Barone, "Toward reliable density functional methods without adjustable parameters: The PBE0 model," J. Chem. Phys. 110, 6158 (1999).