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On Construction of Projection Operators Artur F. Izmaylov J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b01103 • Publication Date (Web): 27 Mar 2019 Downloaded from http://pubs.acs.org on March 28, 2019

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The Journal of Physical Chemistry

On Construction of Projection Operators Artur F. Izmaylov∗ Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario, M1C 1A4, Canada; and Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario, M5S 3H6, Canada E-mail: [email protected]

March 21, 2019 Abstract The problem of construction of projection operators on eigen-subspaces of symmetry operators is considered. This problem arises in many approximate methods for solving time-independent and time-dependent quantum problems, and its solution ensures proper physical symmetries in development of approximate methods. The projector form is sought as a function of symmetry operators and their eigenvalues characterizing the eigen-subspace of interest. This form is obtained in two steps: 1) identification of algebraic structures within a set of symmetry operators (e.g. groups and Lie algebras), and 2) construction of the projection operators for individual symmetry operators. The first step is crucial for efficient projection operator construction because it allows for using information on irreducible representations of the present algebraic structure. The discussed approaches have potential to stimulate further developments of variational approaches for electronic structure of strongly correlated systems and in quantum computing.

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ACS Paragon Plus Environment

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1

Introduction

Projection operators on eigen-subspaces of operators commuting with the Hamiltonian are beneficial for development of various approximate methods of solving both time-independent (TI) and time-dependent (TD) Schrödnger equations (SEs). The operators commuting with the system Hamiltonian are also known as symmetries, and their expectation values are conserved throughout the dynamics. In TI-SE, being able to project the Hamiltonian on one of the symmetries irreducible eigen-subspace ensures that any trial wavefunction that is non-orthogonal to the selected eigen-subspace will have a proper symmetry after the variational procedure and projection. In dynamical problems, approximate Hamiltonians (or Liouvillians) that do not commute with symmetries due to introduced approximations can be symmetrically restored by applying the corresponding projectors, which enforces the approximate dynamics to respect symmetries of the original exact Hamiltonian. The projection becomes crucial if various variable mappings are employed for devising approximations, for example fermionic to qubit Jordan-Wigner 1 or Bravyi-Kitaev 2 transformations, in quantum computing, 3,4 or the mapping between discrete quantum states and continuous variables in quantum-classical dynamics. 5 Very frequently, introducing approximations in the mapped problem violates proper physical symmetries in the original formulation and using the projection operators is a straightforward way to restores the physical symmetries. 6–8 Recently, projection techniques were also extensively used in developing efficient (e.g. polynomial computational cost) methods for treatment of strongly correlated systems. 9–11 In strongly correlated systems, mean-field approaches very frequently undergo spontaneous symmetry breaking. Projections techniques can be used either after variational procedures accounting for electron correlation (projection-after-variation) or within them (variationafter-projection). In both cases, projectors preserve correct physical symmetries in the final wavefunction, moreover, in variation-after-projection, they allow the variational procedure to use the variational flexibility more efficiently. 12 2


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ˆ can be using A naive view on the construction of a projector for a symmetry operator O ˆ |φk i = ok |φk i. Yet, the form Pˆk = |φk i hφk |, where |φk i is a corresponding eigenfunction O this approach is not acceptable because of at least two difficulties: 1) the |φk i hφk | form may not be compactly presentable in the Hilbert space of the problem (e.g. in the coordinate space this operator is non-local integral operator), 2) there can be a degenerate subspace corresponding to a single eigenvalue ok containing potentially infinite number of different terms like |φk i hφk |. Thus, it is more prudent to search for the projection operator in the form ˆ ok ), where F is a function of two arguments. The basic principles of constructing Pˆk = F (O, such functions will be discussed in this work. Another complication in construction of projectors is that, usually, there is not a single ˆ i } commuting with the Hamiltonian, [H, ˆ O ˆ i ] = 0. operator but a whole set of operators {O î, O ˆ j ] 6= 0, and However, in general, these operators do not commute with each other [O thus do not have a common set of eigenfunctions. Therefore, the projection using all known symmetries cannot be simply a projection on an eigen-subspace of a particular operator. The most natural algebraic structure for this set of operators is the Lie algebra. 13,14 This is quite î, O ˆ j ], is again an operator intuitive if we consider that any commutator of two operators, [O ˆ is commuting with the Hamiltonian. Thus, if an initial set of operators commuting with H known, this set can be extended by forming all possible commutators

î, O ˆj ] = [O

X

(k) ˆ cij O k,

(1)

k

(k)

ˆ k } that is closed with respect here cij are some constants. An extended set of operators {O to the commutation operation in Eq. (1) forms a Lie algebra. This Lie algebra will consists of all operators commuting with the Hamiltonian.1 To impose symmetries, the Lie algebraic structure suggests to construct projectors on ˆ n are not defined in the Lie algebra, therefore, it is not possible to build Note that formally powers of O i ˆ i ), such functions would also commute with H ˆ and Taylor series expandable functions for the operators f (O n ˆ could be added to the Lie algebra based on the commutation property. Products Oi are only defined in the universal enveloping Lie algebra. 1

3



irreducible representations of the Lie algebra. This process is well described in mathematical literature on classification of Lie algebras. 13–15 Here, I will provide only a basic summary and examples while referring the interested reader to more specialized literature. One of the essential differences from other works in the literature on symmetry operators will be considering not only irreducible subspaces of Lie algebras but rather construction of projection operators on these irreducible subspaces as functions of the symmetry operators. In what follows I present a theory that allows one to account for algebraic properties of a set of operators commuting with the Hamiltonian and thus to use all available symmetry information to construct projectors on irreducible subspaces of symmetry operators. There are many books written on the subject of application of symmetry in various quantum problems, their main bias is toward application of the group theory where construction of projection operators are well understood. Here, I would like to extend the consideration to Lie algebras that are less frequently discussed in the physical chemistry literature. Even though I will consider a set of operators commuting with the Hamiltonian, the methods discussed here can be applied for a set of operators not necessarily related to symmetries of the Hamiltonian, for example this can be operators commuting with the Liouvillian.

2

Theory

First, treatment of algebraic properties of the symmetry operator set will be considered. ˆ i } form a finite group, Chemists, of course, are more familiar with the situation when {O îO ˆj = O ˆ k , as in the case of point group symmetries. However, I would like to argue that O this group structure is only a useful addition to the more general underlying Lie algebraic ˆ i } form a structure. Nevertheless, it is convenient to separate two cases based on whether {O multiplicative group or not. It will be shown that in the latter (more general case) accounting for the Lie algebraic structure leads to the problem of constructing projection operators for some subset of symmetry operators (e.g. Sˆz , and Sˆ2 for the algebra of spin). Thus, possible

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ways for such constructions will be considered next.

2.1

Projectors for groups

ˆ k } belong to multiplicative group G. Existence of Here, I consider the case when operators {O the group structure allows one to generate projectors on the group irreducible representations following the standard procedure 16 |G|

dΓ X ∗ ˆ ˆ PˆΓ = χ (Ok )Ok , |G| k=1 Γ

(2)

where Γ is the irreducible representation of interest, dΓ is the dimension of Γ, |G| is the ˆ k ) are characters for the group elements. In this case, number of the group elements, and χΓ (O one does not need to deal with Lie algebras and projection operators are simply expressed as a linear function of all operators forming the group.

2.2

Projectors for Lie algebras

In the case when {Oi } only form the Lie algebra, one can obtain a continuous Lie group using the exponentiation of the algebra elements (see appendix A for general exposition). Then the same standard machinery as in the group case can be used for projection construction. This ˆ approach can be illustrated on a simple example of a single symmetry Hermitian operator O. ˆ g(O, ˆ φ) = exp[iφO], ˆ where φ ∈ [0, 2π), allows one to create a continuous Exponentiation of O, ˆ φ). Then the continuous analogue of Eq. (2) can compact cyclic group G with elements g(O, be written as 1 Pˆj = 2π

Z

2π

ˆ

eiφ(O−oj ) dφ,

(3)

0

ˆ All cyclic groups are abelian and have one-dimensional where oj is a particular eigenvalue of O. irreducible representations; each irreducible representation is characterized by the eigenvalue

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oj , hence, the characters of the irreducible representations are exp[iφoj ]. However, switching from the algebra to the group is not necessary to obtain the projectors on the irreducible representations of the algebra. Moreover, historically, Lie algebras were introduced to simplify analysis of the irreducible representations of Lie groups. In any case, understanding irreducible representations of the symmetry Lie algebra is a necessary step for the group pathway (see appendix A) and for a simpler method avoiding the group construction, which is described below. The classification of Lie algebras is a well-developed field 15 and in appendix B it is shown that symmetry operators always form a so-called compact or reductive real Lie algebra. All reductive Lie algebras can be decomposed to a direct sum of abelian and semi-simple sub-algebras. All elements of the abelian part commute with all elements of the algebra, and therefore, the only non-commuting parts that require special consideration is the semisimple sub-algebra. For all semi-simple Lie algebras (e.g. su(2), the electron spin) the standard procedure to construct irreducible representations is to select maximal commuting sub-algebra (i.e. the Cartan sub-algebra), this sub-algebra will form the maximal set of all mutually commuting operators with the Hamiltonian and the corresponding eigen-values represent good quantum numbers, while the eigen-functions form the basis for the irreducible representations. For the well-known su(2)-case, the usual choice of the Cartan sub-algebra is the Sˆz operator. To further characterize the irreducible representations one can use Casimir operators, which commute with all elements of the algebra. By Schur’s lemma this commutativity makes any Casimir operator to be equivalent to the identity multiplied by a constant for any irreducible representation. These constants are eigen-values of Casimir operators on irreducible representations and along with the full set of quantum numbers fully characterize the basis of irreducible representations. In the su(2)-case, Sˆ2 is the Casimir operator an its eigenvalue S(S + 1) along with that for Sˆz , M = −S, ..., S, fully characterize the basis of all irreducible representations. Thus, to construct projectors on the basis states of irreducible representations it is enough to construct projectors on eigenstates of all operators

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of the Cartan sub-algebra and Casimir operators, I will refer to these operators as the fully commuting set. ˆ i in the fully commuting set, individual projectors for eigen-subspaces For each operator O (i) ˆ i and its eigenvalue o(i) determining the Pˆj will be build as a function that depends on O j (i) ˆ i , o(i) ). A total projector on a particular irreducible representation eigen-subspace, Pˆj = F (O j Q ˆ i , o(i) ), where the eigenvalues o(i) should be of the Lie algebra can be written as Pˆ = i Fi (O j j

chosen so that the projectors in the product are not orthogonal to each over (e.g. in the î su(2)-case, M ∈ {−S, ..., S}). The order of the Fi functions does not matter because if O operators commute their eigen-subspace projectors also commute (see appendix C).

2.3

Role of non-commuting elements

Both algebras and groups, generally contain elements that do not commute. It is important to point out a significance of this non-commutativity. It always indicates degeneracy of some eigenstates in the Hamiltonian. In other words, if we characterize some basis functions for irreducible representations using only commuting sub-algebras or sub-groups, non-commuting parts contain information that some of these basis functions form multi-dimensional irreducible representations corresponding to the degenerate eigenstates of the Hamiltonian. If it is not for the non-commuting elements, any Hamiltonian degeneracy would be perceived as completely accidental. Therefore, non-commuting elements providing more symmetry related information on the Hamiltonian spectrum, they are accounted in Eq. (2) for groups and translated through the Casimir operators in algebras.

2.4

Projector as an indicator function of the operator

(i) ˆ i , o(i) ) can be constructed as some differentiable representation For a single projector Pˆj , F (O j

of the Kronecker-delta function. The Kronecker-delta function naturally appears in the spectral decomposition of the projector (here, we consider the non-degenerate case, while the

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degenerate case can be treated similarly with more cumbersome notation) (i) Pˆj =

X

=

X

(i) |φ(i) n i hφn | δnj

(4)

n (i)

(i) , |φ(i) n i hφn | F (x, oj )|x=o(i) n

(5)

n

where we substituted the Kronecker-delta function (also known as an indicator function) with the differentiable function

(i)

F (x, oj ) =

  (i)   1, x = oj ,     (i)

(6)

0, x = on , n 6= j,       ξ(x) ∈ [0, 1], x 6= o(i) n , ∀n,

where ξ(x) can be any smooth function for intermediate values of x. Due to its differentiability ˆ i , o(i) ). Then using we can expand F in the Taylor series, and this expansion can define F (O j (i)

(i) 2

(i)

(i)

the Taylor expansion of F and idempotency |φn i hφn | = |φn i hφn | one can obtain (i) ˆ i , o(i) ). Pˆj = F (O j

(7)

There are multiple ways to define differentiable representation of the Kronecker-delta function (i)

F (x, oj ), here we list several forms: 1) Integration over a unit circle in real space: (i) F (x, oj )

1 = 2π

2π

Z

(i)

eiφ(x−oj ) dφ

(8)

0

(i)

Here, for any x 6= oj we obtain zero. Such selectivity comes with a price of introducing the integral.

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2) Integration over a unit circle in the complex plane: (i) F (x, oj )

1 = 2πi

I

(i)

z (x−oj

−1)

dz

(9)

|z|=1

This function exploits the same idea as the previous one but using the complex plane. 3) The Lagrange interpolation product: (i)

F (x, oj ) =

Y x − o(i) n (i)

n6=j

(i)

oj − on

,

(10)

which is less restrictive since for x-values in between the eigenvalues the functional value is not fixed to zero or one. This polynomial function is used in the Lagrange interpolation ˆ i in Eq. (10) leads to the Löwdin projector operator used method. 17 The substitution, x → O for the spin projection. 18,19 (i)

(i)

4) Integration over an arbitrary contour C(oj ) encircling only z = oj in the complex plane: (i) F (x, oj )

I

1 = 2πi

(i) C(oj )

dz x−z

(11)

ˆ is the resolvent used in investigation of the perturbation series. 20,21 This function with x → H 5) The difference between the limits of logistic functions: (i)

F (x, oj ) = − (i)

(i)

+) −1

(i)

−) −1

lim (1 + e−k(x−oj

k→∞

lim (1 + e−k(x−oj

k→∞

(i)

)

) ,

(12)

where = min |oj − oj±1 |/2. These limits of logistic functions correspond to the Heaviside functions.

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6) “Bump” function (or mollifier):

(i) F (x, oj )

=

   exp

1 (i)

(x−oj )2 −

(i)

1

(i)

1

, x ∈ (oj − 2 , oj + 2 ), (13)

  0, x 6∈ (o(i) − 21 , o(i) + 12 ). j j ˆ i because of the branch choice Unfortunately, this last example cannot be extended to x → O based on x-value in its definition. Equation (10) is especially useful to build projectors for operators with a finite number of eigenvalues because then the product contains a finite number of terms. Interestingly, for such operators, projectors built based on Eqs. (8) and (10) are the same. This is a consequence of only a finite number of linear independent powers for an operator with a finite spectrum. Using the Cayley-Hamilton theorem 22 one can show that any function of such an operator is equivalent to N − 1 polynomial, where N is the number of eigenvalues. Another interesting connection can be found between projectors based on Eq. (8) and generalization of the group projector in Eq. (2) to an infinite continuous one-parametric cyclic group in Eq. (3). In Eq. (4), the spectrum of the symmetry operator is assumed to be discrete, if it is not the case, the Kronecker-delta function needs to be substituted by the Dirac-delta function and its numerous representations as limits of continuous functions.

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2.5

Construction of the indicator function using orthogonality

Another way to present the Kronecker-delta function in Eq. (4) is to build the F function as P an expansion F (x, y) = n fn (x)cn (y) that satisfies the following relations (i) (i) ˆ i , o(i) ) |φ(i) i Pˆj |φk i = F (O j k X (i) ˆ i ) |φ(i) i = cn (oj )fn (O k

(14) (15)

n

X

=

(i)

(i)

(i)

cn (oj )fn (ok ) |φk i

(16)

(i)

(17)

n

X

(i)

(i)

(i)

cn (oj )fn (ok ) = hc(oj )| f (ok )i = δkj .

n

(i)

(i)

Here vectors |f (ok )i and |c(oj )i are defined as (i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

|f (ok )i = {f1 (ok ), f2 (ok ), ...fM (ok )}, |c(oj )i = {c1 (oj ), c2 (oj ), ...cM (oj )}

and are orthonormal for all eigenvalues. The natural question is how to choose fn (x) and cn (y)? One of the choices closely related to the group theory construction of the projectors is to take fk (x) = ck (x) = exp(ikx) and to switch to a continuous version of k with substituting summation by integration ˆ i , o(i) ) = 1 F (O j 2π

Z

2π

(i) ˆ i )dk, c∗k (oj )fk (O

(18)

0

we obtain the projector already introduced in Eq. (3). Note that Eq. (18) is exact for any î, Hermitian operator Oi with a discrete spectrum. Applying further restrictions on operator O one can generate finite sum expansions for the projector on its eigenspaces. Such restrictions are: equidistant separation between neighboring eigenvalues and finite number of eigenvalues. The later condition is less crucial because its violation only leads to projectors that can separate eigen-states within a finite subset. The basic idea of this finite construction is on 11



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the following representation of the Kronecker-delta

δnm

M 1 X 2πik(n−m)/M = e . M k=1

(19)

ˆ i has a finite and equidistant spectrum with the distance between its eigenstates d then If O the projector can be written as

ˆ i , o(i) ) = F (O j

M 1 X 2πik(Oî −o(i) j )/(dM ) e M k=1

M 1 X ∗ (i) ˆ i ), c (o )fk (O = M k=1 k j

(20)

(21)

where ck (x) = fk (x) = exp(2πikx/(dM )). This approach can be used for the electron spin ˆ operators. projection Sˆz and the number of electrons N

3

Conclusions

I reviewed various approaches to construct projectors on irreducible eigen-subspaces of symmetry operators. There are two aspects of this problem: 1) accounting for available algebraic structures of the set of symmetry operators, and 2) construction of individual projection operators as functions of symmetry operators and their eigenvalues. Two algebraic structures, groups and Lie algebras, were discussed. For both structures standard methods of construction of irreducible representations were developed in mathematical literature. Knowledge of irreducible representations helps to construct functions of symmetry operators into the projection operator for a particular irreducible representation. For Lie algebras, various approaches to construct individual projection operators for each symmetry operator were considered. The origin of various projection functions was found in variety of representations for the Kronecker-delta function.

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4

Acknowledgement

A.F.I. is grateful to A. V. Zaitsevskii and V. Y. Chernyak for useful discussions. A.F.I. acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada.

5

Supporting Information

Supporting information contains three appendices further clarifying some aspects of the work. This material is available free of charge via the Internet at http://pubs.acs.org.

References ¨ ¨ das Paulische Aquivalenzverbot. Z. Phys. 1928, 47, 631–651. (1) Jordan, P.; Wigner, E. Uber (2) Seeley, J. T.; Richard, M. J.; Love, P. J. The Bravyi-Kitaev transformation for quantum computation of electronic structure. J. Chem. Phys. 2012, 137, 224109–224109. (3) McArdle, S.; Endo, S.; Aspuru-Guzik, A.; Benjamin, S.; Yuan, X. Quantum computational chemistry. arXiv.org 2018, (4) Olson, J.; Cao, Y.; Romero, J.; Johnson, P.; Dallaire-Demers, P.-L.; Sawaya, N.; Narang, P.; Kivlichan, I.; Wasielewski, M.; Aspuru-Guzik, A. Quantum Information and Computation for Chemistry. arXiv.org 2017, (5) Kim, H.; Nassimi, A.; Kapral, R. Quantum-classical Liouville dynamics in the mapping basis. J. Chem. Phys. 2008, 129, 084102. (6) Ryabinkin, I. G.; Genin, S. N.; Izmaylov, A. F. Constrained Variational Quantum Eigensolver: Quantum Computer Search Engine in the Fock Space. J. Chem. Theory Comput. 2019, 15, 249–255.

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(7) Ryabinkin, I. G.; Genin, S. N.; Izmaylov, A. F. Relation between fermionic and qubit mean fields in the electronic structure problem. J. Chem. Phys. 2018, 149, 214105. (8) Kelly, A.; van Zon, R.; Schofield, J.; Kapral, R. Mapping quantum-classical Liouville equation: Projectors and trajectories. J. Chem. Phys. 2012, 136, 084101. (9) Scuseria, G. E.; Jiménez-Hoyos, C. A.; Henderson, T. M.; Samanta, K.; Ellis, J. K. Projected quasiparticle theory for molecular electronic structure. J. Chem. Phys. 2011, 135, 124108. (10) Qiu, Y.; Henderson, T. M.; Scuseria, G. E. Communication: Projected Hartree Fock theory as a polynomial similarity transformation theory of single excitations. J. Chem. Phys. 2016, 145, 111102. (11) Jake, L. C.; Henderson, T. M.; Scuseria, G. E. Hartree–Fock symmetry breaking around conical intersections. J. Chem. Phys. 2018, 148, 024109. (12) Qiu, Y.; Henderson, T. M.; Zhao, J.; Scuseria, G. E. Projected coupled cluster theory: Optimization of cluster amplitudes in the presence of symmetry projection. J. Chem. Phys. 2018, 149, 164108. (13) Fuchs, J.; Schweigert, C. Symmetries, Lie Algebras, and Representations; Cambridge University Press, 1997. (14) Gilmore, R. Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers and Chemists; Cambridge University Press, 2008. (15) Barut, A. O.; Raczka, R. Theory of Group Representations and Applications; Polish Scientific Publisher, 1980. (16) Wigner, E. P. Group theory and its application to the quantum mechanics of atomic spectra; Academic Press Inc., 1959.

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Page 15 of 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(17) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: Cambridge, 1992. (18) Löwdin, P.-O. Angular Momentum Wavefunctions Constructed by Projector Operators. Rev. Mod. Phys. 1964, 36, 966–976. (19) Löwdin, P.-O. Quantum Theory of Many-Particle Systems. III. Extension of the HartreeFock Scheme to Include Degenerate Systems and Correlation Effects. Phys. Rev. 1955, 97, 1509–1520. (20) Kato, T. Perturbation theory for linear operators; Springer-Verlag, 1966. (21) Messiah, A. Quantum mechanics, Vol II ; North-Holland Publishing Company, 1961. (22) Gel’fand, Izrail’ Moiseevich, Lectures on Linear Algebra; Interscience Publishers, Inc., New York, 1961.

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j ) (i) (i) Z Y 2⇡ on n arbitrary contour )x encircling only z (i) C(oj 1 ˆ i ( O o ) = , F (x, oj ) P j ˆj = (i) e d , (i) ˆ o o Pj | i j n6=j 2⇡ 0 n I dz Y ˆ on dz 1 O (i) ˆj = F (x, oj ) P = , (i) ˆ All 2⇡i z oj jcyclic (i) x n groups particular eigenvalue of a O. C(o )ox z n6 = j ) I j 1 dz ˆ presentations; each representatio = perturbation Pjirreducible ˆ vestigation of the Pj | fori x-values in 2⇡i ˆthe zeigenva ive since between Cj O The Journal of Physical Chemistry

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

...

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r one. This polynomial function is used in th

tions:

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Artur Izmaylov is an Associate Professor at the University of Toronto. He received a Ph.D. with Gustavo Scuseria at Rice University in 2008 and worked as a postdoctoral fellow with John Tully (Yale University) and Michael Frisch (Gaussian Inc.). The main efforts of his group are directed toward developing new computational methods for the electronic structure theory and quantum molecular dynamics on classical and quantum computers.


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On Construction of Projection Operators - The Journal of Physical

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