Comment on "Regularizing the MCTDH equations of motion through an optimal choice on-the-fly (i.e., spawning) of unoccupied single-particle functions" [D. Mendive-Tapia, H.-D. Meyer, J. Chem. Phys. 153, 234114 (2020)]
aa r X i v : . [ qu a n t - ph ] F e b Comment on “Regularizing the MCTDH equations of motion through anoptimal choice on-the-fly (i.e., spawning) of unoccupied single-particlefunctions” [D. Mendive-Tapia, H.-D. Meyer, J. Chem. Phys. 153, 234114(2020)]
Rocco Martinazzo a) and Irene Burghardt b) Department of Chemistry, Universit`a degli Studi di Milano, Via Golgi 19, 20133 Milano,Italy Institute for Physical and Theoretical Chemistry, Goethe University Frankfurt, Max-von-Laue-Str. 7,60438 Frankfurt/Main, Germany (Dated: 25 February 2021)
Time-dependent variational methods are a powerfulapproach to quantum propagation in many dimensions.This is exemplified by the Multi-Configuration Time-Dependent Hartree (MCTDH) method and its hier-archical multi-layer variant, as well as related ap-proaches like Gaussian-based G-MCTDH that connectto a semiclassical trajectory-type picture. These meth-ods rely on time-evolving basis sets whose equations ofmotion are determined by a time-dependent variationalprinciple.
Besides the optimal, variational evolution of a speci-fied basis set, an important problem is the constructionof an adaptive basis where functions are added or re-moved depending on the system’s time-evolving corre-lations. This problem has been typically addressed inan ad hoc fashion, especially in the context of Gaussianwavepacket methods where the notion of spawning dur-ing nonadiabatic events was introduced. In the contextof MCTDH, addition of unoccupied basis functions, no-tably at the start of the propagation, requires regular-ization due to singularities in the equations of motion.To this end, approaches have been proposed which relyon short-time perturbative expansions or natural orbitalpopulation thresholds. In a recent paper, Mendive-Tapia and Meyer presenta modified MCTDH propagation scheme where the ba-sis set is expanded on-the-fly , according to a variationalerror criterion that is augmented by a contribution of ad-ditional, unoccupied basis functions. In the notation ofRef. [11], the augmented error criterion reads˜ E := k ˙Ψ e − ˜˙Ψ M k = min (1)where ˙Ψ e fulfills the Schr¨odinger equation and ˜˙Ψ M is thetime derivative of the augmented MCTDH wavefunction;in the latter, a set of unoccupied basis functions appear via the time derivative of their coefficients and an aug-mented projector. Minimizing the augmented error ofEq. (1) determines the choice of the optimal unoccupied a) Electronic mail: [email protected] b) Electronic mail: [email protected] basis functions in a fully variational setting. The authorsdetail the corresponding algorithm and report superiorperformance of the new scheme as compared with otherspawning approaches, for a model system as well as arealistic system in six dimensions.The purpose of the present Comment is to point outthat the approach of Ref. [11] is an instance of a generalconcept established in previous work where adaptivevariational quantum propagation based on the Local-in-Time Error (LITE) was introduced and exemplified forvariational Gaussian wavepacket dynamics.The LITE, ε M , is defined as the instantaneous devia-tion from the exact solution at the variational minimum, ε M [Ψ ] = 1 ~ min ˙Ψ ∈ T M k i ~ ˙Ψ − H Ψ k (2)where the time derivative of the variational solution, ˙Ψ ,is an element of the tangent space T M of the varia-tional manifold M . The LITE refers to the vari-ationally optimal solution to the short-time dynamicstaking the system from an initial state Ψ to a stateΨ ( dt ) (which generally deviates from the exact solutionΨ exact0 ( dt )). When resizing the variational manifold on-the-fly by spawning, ˙Ψ in Eq. (2) is replaced by ˙Ψ s , andthe modified LITE ε M s [Ψ ] is used to optimally deter-mine the additional basis functions ( M s denotes theextended manifold). This is entirely parallel to the argu-ments given in Ref. [11], and indeed the augmented errorof Eq. (1) can be identified as ˜ E = ε M s [Ψ ].The LITE as defined in Eq. (2) lies at the heart of theMcLachlan variational principle (VP) but it equally ap-plies to the rather common situation where both ˙Ψ and i ˙Ψ lie in T M , entailing that the Dirac-Frenkel VP canbe employed and all the known VPs become equivalent toeach other. The McLachlan VP underscores the ge-ometric idea of optimization by a minimum distance cri-terion and leads to simplified expressions for the squaredLITE, ε M [Ψ ] = 1 ~ ( k H Ψ k − ~ k ˙Ψ k ) (3)and for the error reduction due to spawning, ∆ ε s , whichtakes the form∆ ε s = k ˙Ψ s k − k ˙Ψ k = 1 ~ ℑh δ s ˙Ψ | H | Ψ i (4)Here, δ s ˙Ψ := ˙Ψ s − ˙Ψ is chosen such as to maximize ∆ ε s to achieve optimal spawning. The application of this op-timal spawning criterion was demonstrated in Ref. [12]for variational Gaussian wavepacket dynamics, and anextension to MCTDH was suggested. The connection tothe theoretical developments of Ref. [11] is further de-tailed in App. A.The LITE has various important implications for thevariational dynamics. It determines the a posteriori errorbound for the deviation from the exact solution at time t , k Ψ ( t ) − Ψ exact0 ( t ) k ≤ Z t ε M [Ψ ( τ )] dτ (5)and therefore can be used to minimize the error accumu-lated in time. Furthermore, ε M can be understood asa measure of energy fluctuations accommodated by thevariational manifold. In fact, the LITE is much more than a numerical tool:it is a quantum distance that measures how well a dy-namical approximation performs in the short run. Tosee this, consider the overlap between the variationallyevolved state and the exact state after an infinitesimaltime dt , S ( dt ) = h Ψ ( dt ) | Ψ exact0 ( dt ) i . This overlap canbe connected to a gauge invariant distance D , i.e., theso-called Fubini-Study (FS) distance D (Ψ ( dt ) , Ψ exact0 ( dt )) = 2 (1 − | S ( dt ) | ) (6)which reduces to zero if the variational solution is ex-act. Within the geometric interpretation of quantummechanics, the FS metric is the natural distance in the projective Hilbert space P ( H ) whose elements are physi-cal states that encompass wavefunctions e iφ Ψ that differby an arbitrary phase factor (in practice, the space of thedensity operators for pure states).We will now show, as an extension to Ref. [12], thatthe differential FS distance dD/dt , with D defined in Eq.(6), is identical to the LITE of Eq. (3). To this end, weexpand the overlap up to second order in dt , S ( dt ) = 1 + h ˙Ψ | Ψ i dt − i ~ h Ψ | H | Ψ i dt ++ h ¨Ψ | Ψ i dt − i ~ h ˙Ψ | H | Ψ i dt − ~ h Ψ | H | Ψ i dt + O ( dt ) (7)Here, the first order contribution vanishes due to the sta-tionarity condition with respect to dilations, h ˙Ψ | Ψ i = i ~ − h Ψ | H | Ψ i (Eq. (4) of Ref. [12]). The same equationshows that ℜ h ¨Ψ | Ψ i = ℜ (cid:18) ddt h ˙Ψ | Ψ i − || ˙Ψ || (cid:19) = −|| ˙Ψ || On the other hand, we also have (Eq. (6) of Ref. [12]) ~ || ˙Ψ || = ℑ h ˙Ψ | H | Ψ i All taken together, Eq. (7) reduces to S ( dt ) = 1 + 12 (cid:18) || ˙Ψ || − ~ h Ψ | H | Ψ i (cid:19) dt + O ( dt ) (8)Inserting Eq. (8) into Eq. (6), we find up to second orderin dt , D (Ψ ( dt ) , Ψ exact0 ( dt )) = (cid:18) ~ h Ψ | H | Ψ i − || ˙Ψ || (cid:19) dt = ε M [Ψ ] dt (9)where the term in brackets has been identified as thesquared LITE of Eq. (3). Hence, we obtain ε M = dD/dt .The above derivation also explains the connection withthe energy fluctuations ∆ E = h Ψ | ( H − h H i ) | Ψ i mentioned above, which is made evident when workingin the standard gauge , where Ψ evolves according tothe zero-averaged Hamiltonian, i.e. h Ψ | ˙Ψ +0 i = 0, wherethe superscript + denotes the chosen gauge . This leadsto ε M [Ψ ] = ∆ E ~ − || ˙Ψ +0 || (10)where the energy fluctuation term ∆ E / ~ is known todrive the exact evolution from t = 0 to t = dt D (Ψ (0) , Ψ exact0 ( dt )) = ∆ E ~ dt + O ( dt )As can be inferred from Eq. (10), the variational pathbecomes exact if the squared variational time derivative || ˙Ψ +0 || matches the energy fluctuation term ∆ E / ~ . Inthe language of geometric quantum mechanics, bothterms correspond to magnitudes of velocities in P ( H ),andit is noteworthy that matching these magnitudes is suffi-cient to make the variational solution exact.To return to the spawning criterion of Eq. (4), an ex-pansion of the variational basis in accordance with thiscriterion optimally reduces the mismatch between || ˙Ψ +0 || and ∆ E / ~ in Eq. (10), leading to the largest possiblereduction of the LITE upon extension of the variationalmanifold. In the application shown in Ref. [12], the LITEwas monitored continuously during the propagation, andits value was used to decide when and how to add orremove basis functions. As emphasized in Ref. [12] andunderscored by the above analysis, the LITE providesa “natural” spawning criterion which is firmly rooted inthe variational framework for the approximate solutionto the time-dependent Schr¨odinger equation. We antic-ipate that it will play a crucial role in future on-the-fly variational propagation schemes, bridging between wave-function methods and local, Gaussian wavepacket typebasis sets. Appendix A: Optimal Spawning in MCTDH
For completeness, we show here that the theoretical re-sults of Ref. [11] follow from the general approach devel-oped in Ref. [12], and how they connect to the spawning(“rate”) operator Γ introduced in the latter reference.To this end, we first notice that when the spaces tan-gent to the variational manifold are complex-linear wecan write the difference between the variational and ex-act time derivatives as | ∆ ˙Ψ i = | ˙Ψ i − | ˙Ψ exact0 i = 1 i ~ ( P H | Ψ i − H | Ψ i )= − i ~ Q H | Ψ i (A1)where P is the tangent space projector and Q is its com-plement, Q = I − P . Furthermore, in MCTDH theory —when spawning involves only the κ th degree of freedom— we have for the difference between the variational timederivative with/without spawning | δ s ˙Ψ i ≡ | ˙Ψ s i − | ˙Ψ i = − Ω κ | ∆ ˙Ψ i = 1 i ~ Ω κ Q H | Ψ i (A2)where Ω κ is a projector Ω κ = X α X J | η ( κ ) α Φ ( κ ) J i h η ( κ ) α Φ ( κ ) J | In this expression, the | η ( κ ) α i ’s are the (orthonormal) sin-gle particle functions (spf’s) that are to be added in thespawning process, to be taken from the orthogonal com-plement of the space spanned by the spf’s of the κ th de-gree of freedom. Furthermore, | Φ ( κ ) J i is a configurationexhibiting a “hole” at the κ th position and J is a multi-index running over the occupied spf’s of all degrees offreedom but the κ th (see, e.g., Eq. (22) in Ref. [11]).Upon combining Eq. (A1) and Eq. (A2), we obtain atonce h δ s ˙Ψ | ∆ ˙Ψ i = − ~ h Ψ | H Q Ω κ Q H | Ψ i = −|| δ s ˙Ψ || (A3)This corresponds to the result obtained in Ref. [11] forthe error reduction due to spawning∆ ε s = || ∆ ˙Ψ || − || ∆ ˙Ψ + δ s ˙Ψ || = || δ s ˙Ψ || = 1 ~ h Ψ | H Q Ω κ Q H | Ψ i (A4)On the other hand, from the general theory developedin Ref. [12] (see Eq. (4) of this Comment and, morespecifically, Eq. (12) of Ref. [12] that applies to thepresent context), we find∆ ε s = 1 i ~ h δ s ˙Ψ | H | Ψ i = 1 ~ h Ψ | H Q Ω κ H | Ψ i (A5) which is equivalent to Eq. (A4) provided h Ψ | H Q Ω κ P H | Ψ i = 0i.e., if and only if h δ s ˙Ψ | ˙Ψ i = 0 (A6)since P H | Ψ i = i ~ | ˙Ψ i is the variational equation ofmotion. Now, the condition Eq. (A6) turns out tobe a consequence of the Dirac-Frenkel variational con-dition h δ Ψ | ∆ ˙Ψ i = 0. Indeed, since the latter implies h ˙Ψ | ∆ ˙Ψ i = 0 in both the original and the extendedmanifolds, M and M s , we have0 = h ˙Ψ s | ∆ ˙Ψ s i = h ˙Ψ s | ˙Ψ s − ˙Ψ exact0 i = h ˙Ψ + δ s ˙Ψ | ˙Ψ + δ s ˙Ψ − ˙Ψ exact0 i = h ˙Ψ + δ s ˙Ψ | ∆ ˙Ψ + δ s ˙Ψ i = h δ s ˙Ψ | ∆ ˙Ψ i + h ˙Ψ | δ s ˙Ψ i + || δ s ˙Ψ || ≡ h ˙Ψ | δ s ˙Ψ i where in the last step we have used Eq. (A3). Hence,Eq. (A4) and Eq. (A5) are identical.In Ref. [11], the error reduction is given in the form∆ ε s = 1 ~ h Ψ | H Q Ω κ Q H | Ψ i = 1 ~ X α h η ( κ ) α | ∆ ( κ ) | η ( κ ) α i thereby introducing the single-particle operator∆ ( κ ) = X J h Φ ( κ ) J |Q H | Ψ i h Ψ | H Q| Φ ( κ ) J i that involves the many-body projector Q . The equiva-lence between Eq. (A4) and Eq. (A5) shows, quite re-markably, that one of the two Q projectors is irrelevantfor the error reduction and can be safely omitted.In Ref. [12], we further removed the other Q projectorby requiring additional, simple constraints on the sought-for spf’s, i.e. , forcing them to be orthogonal to both theoccupied space and its time-derivative. Under these con-straints, ∆ ε s ≡ ~ h Ψ | H Ω κ H | Ψ i = 1 ~ X α h η ( κ ) α | Γ ( κ ) | η ( κ ) α i (A7)where Γ ( κ ) = X J h Φ ( κ ) J | H | Ψ i h Ψ | H | Φ ( κ ) J i (A8)is a much simpler (and therefore computationally cheap)generalized “rate” operator, i.e., an effective spawningoperator . Notice that the equality of Eq. (A7) holds strictly under the above conditions, i.e., the approxima-tion lies in a restricted functional variation. As a re-sult, the approximation further provides lower bounds tothe maximum error reduction that can be achieved uponspawning γ ( κ ) α ≤ δ ( κ ) α as follows from the Ritz (Courant-Fischer) theorem whenthe eigenvalues of Γ ( κ ) and ∆ ( κ ) ( γ ( κ ) α and δ ( κ ) α , respec-tively) are sorted in decreasing order of magnitude. H.-D. Meyer, U. 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