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We present a new class of tensor network states that are specificallydesigned to capture the electron correlation of a molecule of arbitrarystructure. In this ansatz, the electronic wave function is represented by aComplete-Graph Tensor Network (CGTN) ansatz which implements an efficientreduction of the number of variational parameters by breaking down thecomplexity of the high-dimensional coefficient tensor of afull-configuration-interaction (FCI) wave function. We demonstrate that CGTNstates approximate ground states of molecules accurately by comparison of theCGTN and FCI expansion coefficients. The CGTN parametrization is not biasedtowards any reference configuration in contrast to many standard quantumchemical methods. This feature allows one to obtain accurate relative energiesbetween CGTN states which is central to molecular physics and chemistry. Wediscuss the implications for quantum chemistry and focus on the spin-stateproblem. Our CGTN approach is applied to the energy splitting of states ofdifferent spin for methylene and the strongly correlated ozone molecule at atransition state structure. The parameters of the tensor network ansatz arevariationally optimized by means of a parallel-tempering Monte Carlo algorithm.

Complete-Graph Tensor Network States: A New Fermionic Wave Function Ansatz for Molecules