### BIVAQUM: an ERC Starting Grant project

Late 2014, Simen Kvaal (PhD) was awarded an ERC Starting Grant for
his proposal BIVAQUM, which is short for *"Bivariational Approximations in Quantum
Mechanics and Applications to Quantum Chemistry."* The project
lasts for 5 years, starting in April 2015.
The
project crosses the borders between **computational and
theoretical chemistry, physics, and applied mathematics.**
The project aims to explore and apply an unconventional variational
principle called **the bivariational principle
(BIVP).** This is a generalization of the standard
(Rayleigh-Ritz) variational principle (VP).
The following is the
abstract from the ERC proposal:

#### Abstract:

The standard variational principles (VPs) are cornerstones of quantum mechanics, and one can hardly overestimate their usefulness as tools for generating approximations to the time- independent and time-dependent Schrödinger equations. The aim of the proposal is to study and apply a generalization of these, the bivariational principles (BIVPs), which arise naturally when one does not assume a priori that the system Hamiltonian is Hermitian. This unconventional approach may have transformative impact on development of ab initio methodology, both for electronic structure and dynamics.

The first objective is to establish the mathematical foundation for the BIVPs. This opens up a whole new axis of method development for ab initio approaches. For instance, it is a largely ignored fact that the popular traditional coupled cluster (TCC) method can be neatly formulated with the BIVPs, and TCC is both polynomially scaling with the number of electrons and size-consistent. No “variational” method enjoys these properties simultaneously, indeed this seems to be incompatible with the standard VPs.

Armed with the BIVPs, the project aims to develop new and understand existing ab initio methods. The second objective is thus a systematic multireference coupled cluster theory (MRCC) based on the BIVPs. This is in itself a novel approach that carries large potential benefits and impact. The third and last objective is an implementation of a new coupled-cluster type method where the orbitals are bivariational parameters. This gives a size-consistent hierarchy of approximations to multiconfiguration Hartree–Fock.

The standard VP is a **cornerstone of quantum
mechanics**, and virtually *all* methods for
approximations are based on it, in some way or another.

*the Schrödinger equation*as an optimization problem. The Schrödinger equation $$ \hat{H}\ket{\psi} = E \ket{\psi} $$ has as solutions precisely those $\psi$ that makes the expectation value functional $\mathscr{E}(\psi)$ an extremum. The functional is defined by $$ \mathscr{E}(\psi) = \frac{\braket{\psi\vert \hat{H}\vert\psi}}{\braket{\psi|\psi}}. $$ The eigenvalue is then given by the value $E = \mathscr{E}(\psi)$ at the extremum. In particular, the ground state is the

*minimizer*. A similar characterization can be made for solutions of the

*time-dependent*Schrödinger equation.

Since the Schrödinger equation is an extremely complicated, it is
essential to be able to devise *well-behaved
approximations*. Since the ground state is the minimum of
$\mathscr{E}(\psi)$, a "variational approximation" is obtained by
inserting a trial function with some parameters and minimizing the
resulting constrained expression. This variational approximation is
always an *upper bound* to the true ground-state energy. The
VP is **mathematically well-understood.**

The **bivariational principle** relaxes the requirement
that $\hat{H}$ be
Hermitian, introducing instead a generalized expectation value
functional
$$ \mathscr{E}(\psi', \psi) =
\frac{\braket{\psi'|\hat{H}|\psi}}{\braket{\psi'|\psi}}. $$
Notably, $\mathscr{E}(\psi',\psi)$ depends on *two*
wavefunctions that are independent, and should be treated on equal footing.
It is straightforward to prove, that the *pair*
$(\psi',\psi)$ is an extremal point of $\mathscr{E}$ if and only if
$\braket{\psi'|\psi}\neq 0$ and
\[ \hat{H}\ket{\psi} = E\ket{\psi} \quad\text{and}\quad
\bra{\psi'}\hat{H} = E\bra{\psi'}. \]
Thus, the Schrödnger equation and its complex conjugate is
satisfied.

The BIVP was introduced indepenently by P.-O. Löwdin and J. Arponen
in 1983. Löwdin studied a non-Hermitian approach to Hartree-Fock
theory. Arponen studied **coupled-cluster theory (CC)**, and
notably, he found that **CC has an elegant formulation using
the BIVP.** This reformulation is considered **highly
unconventional**. This may be due to the fact that the BIVP is
**not mathematically well-understood.** For example,
the functional $\mathscr{E}(\psi',\psi)$ is not bounded-below, so
that devising well-behaved approximations requires more care.

The multireference problem is today perhaps the greatest challenge
in computational chemistry. An important topic for contemporary
research is therefore **multireference coupled-cluster theory
(MRCC)**. Interestingly, there does not seem to be one
natural and unique way to define an MRCC theory. On the contrary, the
current approaches come in many flavors, and all have their
weaknesses and strengths. Current MRCC is **not
based on the BIVP**, and this remains an unexplored path.

After Arponen's initial study, it has **largely been silent
around the BIVP.** In 2012, a novel computational method for
many-electron dynamics based on the BIVP and coupled-cluster theory
was published by Kvaal [*J. Chem. Phys.***136**
194109 (2012)]. This paper forms a
preliminary result for the BIVAQUM project, as it demonstrates
that the BIVP is useful also outside the standard coupled-cluster
context. It is therefore natural to ask the following:

#### Research question:

Can the **bivariational principle** and the
corresponding approximations be put on a firm mathematical ground?
Can the BIVPs be used as general principles to develop new and
understand existing ab initio wavefunction methods for quantum
chemistry, both for electronic structure and dynamics?

The BIVAQUM project is of interdisciplinary character. The BIVP must
be studied from a **mathematical perspective**,
involving spectral theory of linear operators and functional
analysis. Furthermore, the archetypical bivariational method is
the **traditional coupled-cluster method** — the
gold-standard of quantum chemistry. Moreover, **multireference
coupled-cluster theory** lacks a foundation using a
variational principle, and BIVAQUM aims to study the multireference
problem from the perspective of the BIVP. Finally, in contemporary
nuclear physics, CC methods are among the most important tools.

See also the literature list for suggested reading.