UC Berkeley

The interplay between mathematics and physics has a long and rich history. Recently, ideas from algebraic geometry have played an increasingly important role in the study of physical phenomena, including those arising in particle physics, quantum mechanics, and cosmology. This exchange has led to major advances in both fields and continues to open new directions. In this thesis, I establish a novel connection between algebraic geometry and quantum chemistry. Through methods from nonlinear algebra, with particular emphasis on combinatorics, and representation theory, I develop geometric formulations of coupled cluster theory that lead to new structural, enumerative, and computational results.First, we develop an algebraic-geometric framework for coupled cluster (CC) theory. At the heart of quantum chemistry is the problem of solving the electronic Schrödinger equation, which can be formulated as a finite but high-dimensional eigenvalue problem. To study this problem, we introduce the truncation varieties, a family of projective varieties parameterized by Laplace polynomials. These generalize the Grassmannian in its Plücker embedding. We then approximate this eigenvalue problem by a lower-dimensional nonlinear eigenvalue problem on the truncation varieties. This leads to a hierarchy of polynomial systems of equations, known as the unlinked coupled cluster equations. We define the coupled cluster degree, an invariant of the truncation varieties, as the generic number of solutions to these equations. By relating this degree to the total degree of a graph, we derive an explicit formula for the CC degree of the Grassmannian of lines. Using toric degenerations we also relate the CC degree of the Grassmannian with the volume of a polytope. Together with numerical algebraic methods, this algebraic framework enables us to completely solve main variants of the CC equations for the molecules LiH and H4.Next, we develop a second-quantized framework for coupled cluster theory, in which quantum states and observables are expressed in terms of polynomials in operators. To this end, we introduce the Fermi–Dirac algebra, the Clifford algebra generated by the creation and annihilation operators acting on the fermionic Fock space F ∼= ∧R n . We describe a Gröbner basis for this algebra and thereby obtain an alternative proof of Wick’s theorem, a fundamental result in second quantization. We then realize the Hamiltonian as an element of the Fermi–Dirac algebra. Within this framework, we reformulate the truncation varieties and the coupled cluster equations in second quantization. By dropping the assumption of particle conservation, we obtain an extended family of truncation varieties, which includes many well-known varieties, such as flag varieties and the spinor variety.Finally, we turn to spin, which is encoded by an SU(2)–action on the quantum states. We show that the dimension of the SU(2)–invariant subspace of the state space is given by the Narayana numbers, a refinement of the Catalan numbers. In quantum chemistry, this subspace is called the spin singlet sector and consists of states with total spin zero. We identify the spin singlet sector with a commutative ring defined by cubic relations. This identification, together with the RSK correspondence, yields an explicit bijection between the basis states of the spin singlet sector and the Dyck paths counted by the Narayana numbers. Because the Hamiltonian is SU(2)–invariant, we may restrict to the spin singlet sector and hence to the corresponding spin-adapted truncation varieties. We show that the Veronese square of the Grassmannian appears as a spin-adapted truncation variety. Compared with the spin-generalized formulation, the spin-adapted approach yields a substantial reduction in both dimension and degree, reducing the coupled cluster degree by orders of magnitude. We present scaling studies showing these asymptotic improvements. We exploit this reduction to compute the full solution landscapes of spin-adapted CC equations for LiH and water, showing that spin symmetry makes previously intractable systems accessible to algebraic methods.