Yang Zhang

Peifeng Su

Benjamin Lasorne

Benoît Braïda

Wei Wu

A Novel Valence Bond based Automatic Diabatisation Method by Compression

A novel valence bond based automatic diabatisation method by compression, called valence bond based compression approach for dibatization (VBCAD), is presented in this letter. It is a "blackbox" type method that provides an automatic diabatisation from a classical valence bond (VB) perspective. In VBCAD, a model space projection is performed by an eigenvalue decomposition algorithm followed by dimensional reduction based on a sequence of Householder transformations.

Our diabaticity criterion is implemented in a way that maximizes the diversity of VB structure weights between different diabatic states. Owing to the rigorous Householder transformations employed in this entire procedure, the invariance of the target eigensubspace is preserved. This is illustrated on two prototypical examples.

Table of Content (TOC)

It is well established that the Born-Oppenheimer (B.O.) approximation breaks down near conical intersections or avoided crossings, where at least two adiabatic states are degenerate or nearly degenerate. As a consequence, the molecular Hamiltonian must account for non-adiabatic coupling terms (NACT) among more than one adiabatic states, which still represents a challenge for its implementation within quantum dynamics simulations. The typical procedure relies on a unitary change of electronic basis, the so-called diabatic states, which are defined so as to reduce the magnitude of NACTs and make them negligible in molecular Hamiltonians. As such, the singular kinetic NACT vectors can be replaced by coupled potential energy surfaces that are smoothly varying functions of the nuclear coordinates. In principle, the construction of approximate diabatic states is not unique. The direct way that stems from their formal definition requires the determination of all derivative couplings over an extended range of nuclear coordinates, which involves large computational efforts [START_REF] Smith | Diabatic and Adiabatic Representations for Atomic Collision Problems[END_REF][START_REF] Baer | Adiabatic and Diabatic Representations for Atom-Molecule Collisions: Treatment of the Collinear Arrangement[END_REF][START_REF] Top | Incorporation of Electronically Nonadiabatic Effects into Bimolecular Reactive Systems. I. Theory[END_REF][START_REF] Baer | Electronic Non-Adiabatic Transitions Derivation of the General Adiabatic-Diabatic Transformation Matrix[END_REF] .

Therefore, some alternative strategies have been proposed over the years, based on the properties of the adiabatic wave functions, whereby the adiabatic-to-diabatic (ATD) transformation is defined in order to enforce the smoothness of physical properties [START_REF] Mulliken | Molecular Compounds and Their Spectra[END_REF][START_REF] Werner | MCSCF Study of the Avoided Curve Crossing of the Two Lowest1σ+States of LiF[END_REF][START_REF] Hush | Intervalence-Transfer Absorption.Ⅱ. Theoretical Considerations and Spectroscopic Data[END_REF][START_REF] Cave | Calculation of Electronic Coupling Matrix Elements for Ground and Excited State Electron Transfer Reactions: Comparison of the Generalized Mulliken-Hush and Block Diagonalization Methods[END_REF][START_REF] Ren | Two-Dimensional Analysis of the Diabatic Transition of a General Vectorial Physical Observable Based on Adiabatic-to-Diabatic Transformation[END_REF] , or of the expressions of the electronic wave functions in terms of configurations [START_REF] Hendekovic | Novel Variational Definition of Diabatic States[END_REF][START_REF] Ruedenberg | A Quantum Chemical Determination of Diabatic States[END_REF][START_REF] Atchity | Determination of Diabatic States through Enforcement of Configurational Uniformity[END_REF][START_REF] Nakamura | The Direct Calculation of Diabatic States Based on Configurational Uniformity[END_REF][START_REF] Spiegelmann | The Use of Effective Hamiltonians for the Treatment of Avoided Crossings. I. Adiabatic Potential Curves[END_REF][START_REF] Spiegelmann | The Use of Effective Hamiltonians for the Treatment of Avoided Crossings. Ii. Nearly Diabatic Potential Curves[END_REF][START_REF] Li | Model Space Diabatization for Quantum Photochemistry[END_REF] . As large non-adiabatic couplings are related to fast changes of the adiabatic wave functions with respect to the nuclear coordinates, the key of these strategies is to construct the ATD transformation from its ability to reduce the changes. In particular, the diabatic states can be obtained from the concept of block-diagonalization of the Hamiltonian [START_REF] Pacher | Approximately Diabatic States from Block Diagonalization of the Electronic Hamiltonian[END_REF][START_REF] Cederbaum | Block Diagonalisation of Hermitian Matrices[END_REF][START_REF] Pacher | Quasidiabatic States from Abinitio Calculations by Block Diagonalization of the Electronic Hamiltonian: Use of Frozen Orbitals[END_REF] , which can be derived from a least action principle.

In classical VB theory, the wave function of a many-electron system can be generally expressed as a linear combination of VB structures 20 ,

(

) 1
where CK is the coefficient of VB structure FK. In classical VB theory, the VB structures are constructed by using VB orbitals that are strictly localized on an atom or fragment. To do so, each VB orbital is defined as a linear combination of a subset of the basis functions, including the basis functions

K K K C Y = F å
centered only on a chosen atom or fragment. Among various classical VB methods, VBSCF, which optimizes structure coefficients and orbitals simultaneously, is the basic method of ab initio classical VB theory. Each VB structure is defined by a particular configuration of electrons (understood as a given occupancy of the VB orbital set) together with a specific spin function. As such, each VB structure corresponds to a particular bonding scheme that is kept invariant irrespective of the variations of molecular geometries, and, can be mapped to a specific Lewis structure. Therefore, classical VB theory appears as a natural choice from a chemical perspective in the construction of diabatic states.

Recently, ab initio classical VB methods and block-localized wave function (BLW), which is a type of semi-localized VB method that uses a single-determinantal wave function built on semi-localized molecular orbitals, have been successfully employed in the direct construction of diabatic states, based on the assumption that an educated inspection of dominant VB structure sets can be associated with the diabatic states [START_REF] Grofe | Diabatic-at-Construction Method for Diabatic and Adiabatic Ground and Excited States Based on Multistate Density Functional Theory[END_REF][START_REF] Song | On the Construction of Diabatic and Adiabatic Potential Energy Surfaces Based on Ab Initio Valence Bond Theory[END_REF][START_REF] Lin | Explicit Construction of Diabatic State and Its Application to the Direct Evaluation of Electronic Coupling[END_REF] .

In this letter, we go beyond this assumption and present a novel method named valence bond based compression approach for dibatization (VBCAD). In the VBCAD scheme, diabatic states are obtained by a sequence of rigorous unitary transformations that keeps the invariance of the target eigensubspace. The whole procedure is schematically presented in the flowchart displayed in Figure 1. The central idea of VBCAD is to employ first a projection to reduce the full Hamiltonian to a low rank matrix that preserves the target eigensubspace. This is followed by a sequence of Householder transformations (prediagonalisation) coupled to a VB based diabatisation criterion that produces the diabatic low rank and low dimension block of interest. The whole procedure ensures smoothness of the final diabatic matrix elements with respect to variations in the nuclear coordinates (molecular geometry). The different steps of the VBCAD procedure are detailed in the following.

(Figure 1. near here)

In order to facilitate the following eigenvalue decomposition (ED) formalism with the non-orthogonal symmetric Löwdin orthogonalization [START_REF] Löwdin | On the Nonorthogonality Problem[END_REF][START_REF] Löwdin | On the Non-Orthogonality Problem Connected with the Use of Atomic Wave Functions in the Theory of Molecules and Crystals[END_REF] , i.e.:

(

where H and M are the VB Hamiltonian and overlap matrices, respectively.

The rank reduction trick starts from ED, which is a factorization of the form,

where: i) Q is an orthogonal matrix (QQ

where and denote the diagonal matrices with eigenvalues , ..., and , ..., respectively, with r the number of electronic states that are required to describe the system under study over a representative range of nuclear coordinates (i.e.: r is the dimension of the target space). The feature of the projected ED approach is that the model subspace is isolated. This means that the following matrix compression will be performed on H e with reduced rank r rather than on H L with full rank n.

A series of Householder matrices are then used to achieve the dimensional reduction in the proposed method. In this sense, the symmetric high dimension (n×n) matrix H e can be reduced into a low dimension (r×r) matrix H pre-dia , which is termed "pre-diabatic":

(

H pre-dia is obtained by the following recursion formula:

1 1 L T 2 2 - - H = (M ) H(M ) L -1 H = QΣQ Σ 1 e n e e e n n ae ö ae ö ç ÷ ç ÷ è ø è ø L -1 -1 0 0 Σ 0 H = Q Q +Q Q = H + H 0 Σ 0 0 e Σ n Σ 1 e r e +1 r e n e pre-dia 1 e - H =P H P (i=1, 2, … , n-1) (6)
in which the i th transformation Pi is defined as: (7) where is a column unit vector related to the matrix elements in . The detailed definition of is provided in Appendix.

Sequentially, the VB structure coefficients of pre-diabatic wave functions are obtained by the relations:

(8)
where C e is a coefficient matrix of dimension (n×r) with the eigenvector of on each column respectively, and comes from the symmetric Löwdin orthogonalization. It should be noticed here that this Householder reduction algorithm is feasible for any number of states of interest.

The final transformation from the pre-diabatic Hamiltonian to the diabatic Hamiltonian is achieved by the following unitary transformation:

For the special case r=2, the unitary matrix U can be defined as a two-step Householder transformation with two variables, by the relation: (10) where U1(a) and U2(b) are Householder matrices defined by eq.7, in which w1 = (sina, cosa,0,…,0) T ; w2 = (sinb, cosb,0,…,0) T

Now the key point becomes how to characterize the diabatic states. This means that a specific diabaticity criterion is required so as to act as a constraint to make eq.9 nonarbitrary. Taking advantage of the characteristics of classical VB structures, principally their direct and unchanged connection with e e 1 i i i i

+ = H PH P T 2 i i i w w = - P I i w e i H i w 1 pre-dia e 2 - = C M PC e Σ 1 2 - M 1 pre dia dia - - = U H U H ( ) ( ) ( ) 1 2
,

a b a b = U U U
specific Lewis structures, we propose here a novel approach, whereby the diabaticity criterion is chosen as a quantity that maximizes the separation of VB structure weights in different states. This criterion was implemented in VBCAD to obtain the diabatic states in such a way that, in eq.9,

is determined by maximizing the quantity defined as: (12) where n is the number of VB structure, 1 and 2 correspond to the two diabatic states, WK,i is the weight of structure K in the diabatic state i. Among various available definitions of structure weights, the simplest renormalized definition of VB weights was employed here. Within this definition, the VB structure weights WK are obtained as:

where is the coefficient of structure K in a given diabatic state.

This particular choice of will be referred to as max in the following. Therefore, the diabatic Hamiltonian H dia is obtained from eq.9:

(

The VB coefficients of both diabatic states, which are denoted as , are obtained from:

Subsequently, the attribution of pointwise diabatic states can be determined according to the minimum change in coefficients.

Two examples, the torsion of ethylene (C2H4) and the photo-dissociation of formaldehyde (H2CO), are presented to illustrate our method. For the torsion of ethylene the adiabatic states were calculated by the VBSCF method with six active electrons in six active orbitals. The strictly-localized valence orbitals were recombined in symmetry-adapted semi-localized orbitals to match the symmetry of the (Figure 2. near here)

( , ) a b U [ ( , )] F a b U ,1 ,2 [ ( , )] ( , ) ( , ) n K K K F W W a b a b a b = - å U 2 2 1 ; n K K K K W NC N C = = å K C ( , ) a b U ( , ) a b U 1 pre dia dia max max ( , ) ( ,
Note that the first A state is the ground state (N state), while the second A state is essentially the doubly excited state for the ππ* system (Z state). As expected, our results show that the two corresponding adiabatic energies exhibit an avoided crossing around the perpendicular geometry, while the character of the two states switches along the pathway. This is reflected by the crossing of the H11 and H22 curves.

The two diabatic states thus transform smoothly with the variation of the dihedral angle. Furthermore, the VBCAD scheme deals correctly with the region of the avoided crossing, in the middle of the reaction path. This means that the numerical values of H12, which have the meaning of a "diabatic coupling", vary little with respect to the central value that can be calculated directly when H11 = H22, (16) where E1 and E2 are the energies of the adiabatic states. Further from the avoided crossing region the diabatic coupling term, H12, varies smoothly and decays to nearly zero at the planar geometries.

In order to estimate the variance between pre-diabatic and diabatic states, the overlaps of the wave functions, , are computed pointwise, as shown in Figure 3. It comes out that the average values of the overlaps are close to 1 along the pathway, which means that the "pre-diabatic" states are already good approximations of the diabatic states that were obtained by further maximizing the difference of composition of the wave functions in terms of VB structures.

(Figure 3. near here)

Let us now move to our second numerical example. The photoexcitation of formaldehyde can ( )

12 2 1 1 2 H E E = - pre-dia dia Y Y
lead either to the loss of H2 (molecular dissociation), or to the loss of H (radical dissociation) [START_REF] Araujo | The Photochemistry of Formaldehyde: Internal Conversion and Molecular Dissociation in a Single Step?[END_REF] . The non-adiabatic reaction path for the H2 + CO molecular dissociation along a CS symmetry-preserving pathway is explored here. The six electronic states that should be considered in a complete treatment of this dissociation pathway are displayed in Figure 3, and will be referred to as S0-S6 in the following.

As can be seen on the figure, an avoided crossing occurs between the S0 and S2 states of A' symmetry, which will be a particularly interesting region to test the efficiency of our VBCAD methodology.

An accurate treatment of this dissociation pathway requires an active space of eight electrons into seven active orbitals, leading to a set of 490 VB structures on which the different states expand (eq 1).

The full VBSCF adiabatic energy curves of the six states are shown in (Figures.4, 5 and 6 near here)

In order to further verify the diabaticity of states obtained by VBCAD, the residual non-adiabatic coupling (RNAC) is evaluated numerically. The RNAC is expressed as:

where α and β are the labels of the two states, Q is the set of nuclear coordinates, K is the index of the degree of freedom. For the states to be (quasi) diabatic, the non-adiabatic coupling must satisfy the following criterion:

Thus, evaluating the overlap between the two states along a pathway will give a measure of diabaticity: it has to be near for the non-adiabatic coupling to be small. Since the diabaticity of each VB structure is maintained by the electron-pairing pattern, and it varies little with the variation of the nuclear coordinates, the overlap between the two states along a pathway can be approximately obtained by the "dot product" of VB structure coefficients of the two states, which becomes a feasible way to evaluate the RNAC. The numerical results are shown in

K K K K K K Q D Q Q b a b a b a a b a b d d d d ¶Y Y Y + » Y Y + Y = + ¶ å å Q Q Q Q Q Q Q Q ( ) 0 K D Q ab » ab d
This is also a pointwise method which thus can be turned into a global diabatisation technique without line-integration along a path. In VBCAD, the model space projection is implemented by an eigenvalue decomposition (ED) algorithm followed by a sequence of Householder compressions, then the diabaticity criterion that makes use of maximally separated VB character is imposed through a final Householder transformation step. This diabatisation procedure keeps the invariance of the subspace spanned by each eigenstate of interest during the compression. It is appealing that the automation in VBCAD strategy provides a practical way of automatic inspection of VB structures based on the contributions of each diabatic state. The examples shown here illustrate that there is a real possibility for ab initio VB calculations to achieve the production of diabatic states by Householder matrix compression. Finally, it should be noted that we did not consider dynamic correlation here. VBSCF calculations for strongly correlated systems may not be accurate enough, and can be improved by using post-VBSCF methods [START_REF] Wu | Valence Bond Configuration Interaction: A Practical Ab Initio Valence Bond Method That Incorporates Dynamic Correlation[END_REF][START_REF] Chen | A Valence Bond Method That Incorporates Perturbation Theory[END_REF][START_REF] Ying | A Density-Functional-Based Valence Bond Method[END_REF][START_REF] Zhou | Hamiltonian Matrix Correction Based Density Functional Valence Bond Method[END_REF][START_REF] Ying | A Valence Bond Based Multi-Configurational Density Functional Theory with a Single Variable Hybrid Parameter[END_REF] . This will be considered and developed in the future, as well as the generalization of the algorithm used for VBCAD to cases with more than two states.

The computational details are summarized as follows: The full Hamiltonian and overlap matrices were calculated with the Xiamen Valence Bond (XMVB) program [START_REF] Chen | XMVB 2.0: A New Version of Xiamen Valence Bond Program[END_REF] . The reaction path of formaldehyde was obtained with the GAUSSIAN16 software 33 . The basis set 6-31G* was used in all calculations. The diabatisation procedure was performed by a post-processing algorithm with the MATLAB program version 2018b 34 . with rank r can be reduced to a (r+1)×(r+1) tridiagonal matrix. In the next step, it can be further transformed perfectly into a smaller r×r full rank submatrix by means of a two-side Householder compressor Pn-1. Here we define u as a column vector corresponding to the one-dimensional null-space after tri-diagonalization of H e , ui is the i-th element, expressed by

T 1 i i w w = e i H i s 1, e i i n i n i i G H s s -+ - = + i w 1, , 2 e n i n i i i n i i H a G s -+ - - + = (A-7)
The (n-1) th transformation in eq.6 defined by eq.7 acts as a compressor, in which wn-1 is computed by the relations, ; ;

x y R R Y Y

  at all twisted geometries. The dihedral angle ÐH-C-C-H was varied from 0° to 180°, while the bond lengths of the four C-H bonds and C-C bond were fixed at 1.070 Å and 1.355 Å respectively. The energies of the lowest two adiabatic states of A symmetry together with the diabatic energy curves obtained through the VBCAD procedure are presented in Figure.2.

Figure 4 . 5 .

 45 From the adiabatic curves we can see that two adiabatic states exhibit an S0/S2 avoided crossing along the reaction coordinate around point 10. It should be noted here that around the dissociation region, the excited states are highly degenerate and a more detailed analysis of them is beyond the scope of this paper. As a consequence, the region around the S0/S2 avoided crossing (from point 9 to point 12 in Figure 4.) can be simplified as a two-state case. The results are presented in Figure The absolute value of the off-diagonal element does not vary much as a function of the reaction coordinate. This is indicative of the diabatic character of the representation produced by VBCAD. As can be seen from Figures 6 (a)and (b), in contrast with the sudden change in the electronic nature of the adiabatic states, the diabatic states preserve their respective natures and are mainly governed by VB structure groups with specific electron occupation: one (green curve) is dominated by a group of VB structures with a doublyoccupied lone-pair on the O atom with a single electron on C(σ), while the other (red curve) is dominated by that of a singly-occupied lone-pair on the O atom with two electrons on C(σ). The analysis from Figure6(c) suggests a possible electron transfer between the O and C atoms around the avoided crossing region. Such information on the chemical nature of the diabatic states was obtained automatically through VBCAD but was not so obvious a priori from a simple inspection of VB structures. This illustrates the capabilities of our approach in situations where diabatic states are not obvious from the onset.

(A- 8 )

 8 With these choices of n-1 Householder matrices above, H e becomes a full-rank matrix H pre-dia with rank r.

Figure 1 .

 1 Figure 1. Flowchart representing the different steps of the VBCAD framework, with n the full dimensionality of the problem into study, and r the dimensionality of the reduced target space.

Figure 2 .

 2 Figure 2. Black continuous curves: energy evolution (E1 and E2) of the lowest two adiabatic A states of C2H4 along the torsion coordinate describing a D2-symmetry preserving pathway; red and green dashed lines: the diagonal diabatic energies H11 and H22 as obtained through the VBCAD procedure; blue dotted line: the diabatic coupling term H12.

Figure 3 .

 3 Figure 3. Wave function overlap between pre-diabatic and diabatic states along the distortion pathway.

Figure 4 .

 4 Figure 4. Energy curves for the photodissociation of formaldehyde along a Cs pathway. Black lines correspond to the three lowest lying A' states, blue lines to the three lowest lying A" states.

Figure 5 .

 5 Figure 5. VBCAD results for the photodissociation of formaldehyde along a Cs pathway around the avoided crossing region. The black lines correspond to the two lowest lying A' adiabatic states. Green

Figure 6 .

 6 Figure 6. The specific VB structure group weights of two diabatic states around the avoided crossing region. The curve with red and green dots denotes the H11 and H22 diabatic states respectively. Y-axis displays the sum of weights of VB structures that describe the electron occupation on: (a) C(σ) orbital; (b) O(n) orbital; (c) both C(σ) orbital and O(n) orbital.

Table . 1

 . , which suggest that the values of RNAC are small, as expected for genuine diabatic states.In summary, a novel valence bond based automatic diabatisation method by compression, referred to as VBCAD, is presented. It allows a low-dimensional effective Hamiltonian to be built, based on a diabatic VB expansion. VBCAD is "black-box" with no parameter involved in the whole procedure.

	;	;	;	;	;	;	( )
				(Table.1 near here)

Table 1 .

 1 The values for the two diabatic states; Rx and Ry denote successive points along the reaction coordinate.

	Y	dia	;	x R	Y	dia	;	y R	<9|9.5>	<9.5|10>	<10|10.5>	<10.5|11>	<11|11.5>
			Diab_a			0.9940	0.9951	0.9754	0.9548	0.9762
			Diab_b			0.9989	0.9982	0.9970	0.9947	0.9958
							dia	dia
							1	2

Acknowledgement:

This project is supported by the National Natural Science Foundation of China (No. 21733008), New Century Excellent Talents in Fujian Province University and the Fundamental Research Funds for the Central Universities (No. 20720190046).

Here we introduce the Householder compression in a detailed way. In the main body of the paper, we focus on a special case with r=2 as an illustration, in which the Householder tri-diagonalization [START_REF] Wilkinson | Householder's Method for Symmetric Matrices[END_REF] and Householder compressor [START_REF] Strobach | The Householder Compressor Theorem and Its Application in Subspace Tracking[END_REF] algorithms are well suited for the proposed method.

For the 1 st to (n-2) th step of transformation in eq.5,

wi = [ai,1; ai,2;…; ai,n-i; 0; …; 0] T (A-1)

where wi is a column vector related to the matrix elements in , with the non-zero coefficients ai,k defined in the following. Because the transformation with Pi (eq.6) leaves the last (i-1) row and column unaltered, hence the last several elements of wi are chosen to be zero.


Csaba Fábri

Benjamin Lasorne

Gabor J Halasz

Lorenz S Cederbaum

Ágnes Vibók
email: vibok@phys.unideb.hu

Gábor J Halász

Striking Generic Impact of Light-Induced Non-Adiabaticity in Polyatomic Molecules

Keywords: Light-induced conical intersection, Vibrational spectrum, Pump-probe scheme, Intensityborrowing effect

come   L'archive ouverte pluridisciplinaire

decades to play a fundamental role in most photobiological, photochemical, and photophysical processes. [START_REF] Lim | Experimental probing of conical intersection dynamics in the photodissociation of thioanisole[END_REF][START_REF] Wörner | Conical Intersection Dynamics in NO 2 Probed by Homodyne High-Harmonic Spectroscopy[END_REF][START_REF] You | ππ * /πσ * ) Conical Intersection Seam Experimentally Observed in the S-D Bond Dissociation Reaction of Thiophenol-d 1[END_REF][START_REF] Musser | Evidence for conical intersection dynamics mediating ultrafast singlet exciton fission[END_REF][START_REF] Von Conta | Conical-intersection dynamics and ground-state chemistry probed by extremeultraviolet time-resolved photoelectron spectroscopy[END_REF][START_REF] Corrales | Coulomb Explosion Imaging for the Visualization of a Conical Intersection[END_REF][START_REF] Martinez | Seaming is believing[END_REF][START_REF] Kowalewski | Catching Conical Intersections in the Act: Monitoring Transient Electronic Coherences by Attosecond Stimulated X-Ray Raman Signals[END_REF][START_REF] Xie | Nonadiabatic Tunneling in Photodissociation of Phenol[END_REF][START_REF] Devine | Non-Adiabatic Effects on Excited States of Vinylidene Observed with Slow Photoelectron Velocity-Map Imaging[END_REF][START_REF] Woo | Real-Time Observation of Nonadiabatic Bifurcation Dynamics at a Conical Intersection[END_REF][START_REF] Curchod | Ab Initio Nonadiabatic Quantum Molecular Dynamics[END_REF][START_REF] Bennett | Monitoring molecular nonadiabatic dynamics with femtosecond X-ray diffraction[END_REF][START_REF] Ryabinkin | Geometric Phase Effects in Nonadiabatic Dynamics near Conical Intersections[END_REF][START_REF] Xie | Up to a Sign. The Insidious Effects of Energetically Inaccessible Conical Intersections on Unimolecular Reactions[END_REF] Such effects become significant around molecular geometries where two (or more) potential energy surfaces (PESs) are energetically degenerate, such that the Born-Oppenheimer (BO) approximation [START_REF] Born | Zur Quantentheorie der Molekeln[END_REF] breaks down. This yields strong mixing among BO electronic states and subsequent radiationless transitions correlated with variations of the molecular geometry (nuclear degrees of freedom).

Such degeneracy points, termed conical intersections (CIs), are characterized locally by a two-dimensional subspace within the space spanned by the nuclear degrees of freedom, the so-called "branching" space along which the degeneracy is lifted to first order around the CI. This can occur only in molecules with more than two atoms (Wigner's noncrossing rule). [START_REF] Von Neumann | Über das Verhalten von Eigenwerten bei adiabatischen Prozessen[END_REF] Indeed, diatomic molecules have a single vibrational coordinate, which cannot accommodate both zero energy-difference tuning and zero coupling except for symmetry reasons. CIs provide ultrafast nonradiative decay channels for electronically excited molecules. [START_REF] Coe | Competitive Decay at Two-and Three-State Conical Intersections in Excited-State Intramolecular Proton Transfer[END_REF][START_REF] Boggio-Pasqua | Ultrafast Deactivation Channel for Thymine Dimerization[END_REF] In the vicinity of such points, nonadiabatic dynamics (beyond BO) takes place typically on the femtosecond time scale (∼10-100 fs). While it is by now well established that they are present in small-or medium-sized molecules, CIs have been shown to be ubiquitous in large molecular systems, hence playing a key role in most photoinduced processes. CIs can also influence the ground-state spectroscopic and dynamical properties of molecules. [START_REF] Matsika | Conical Intersections of Three Electronic States Affect the Ground State of Radical Species with Little or No Symmetry: Pyrazolyl[END_REF][START_REF] Xie | Signatures of a Conical Intersection in Adiabatic Dissociation on the Ground Electronic State[END_REF] More than a decade ago, it has been demonstrated that laser light with significant intensity (dressed-state regime) can in turn induce CI-type situations between field-dressed PESs and thus give rise to new types of nonadiabatic phenomena, even in diatomic molecules. [START_REF] Moiseyev | Laser-induced conical intersections in molecular optical lattices[END_REF][START_REF] Šindelka | Strong impact of light-induced conical intersections on the spectrum of diatomic molecules[END_REF] Such points were then termed light-induced conical intersections (LICIs). Whereas the location of a "natural" CI in a field-free polyatomic molecule and the strength of the nonadiabatic coupling are inherent properties of the system, the occurrence of the LICI is determined by the laser frequency while the strength of the nonadiabatic-type coupling is controlled by the laser intensity.

Several theoretical and experimental works have demonstrated on both fronts that LICIs

give rise to a variety of unexpected nonadiabatic phenomena in molecules, [START_REF] Halász | Conical intersections induced by light: Berry phase and wavepacket dynamics[END_REF][START_REF] Halász | Light-Induced Conical Intersections: Topological Phase, Wave Packet Dynamics, and Molecular Alignment[END_REF][START_REF] Halász | Nuclear-wave-packet quantum interference in the intense laser dissociation of the D + 2 molecule[END_REF][START_REF] Halász | Geometric phase of light-induced conical intersections: adiabatic time-dependent approach[END_REF][START_REF] Natan | Observation of Quantum Interferences via Light-Induced Conical Intersections in Diatomic Molecules[END_REF] with significant impact on various dynamical [START_REF] Halász | Direct Signature of Light-Induced Conical Intersections in Diatomics[END_REF][START_REF] Csehi | Towards controlling the dissociation probability by light-induced conical intersections[END_REF][START_REF] Csehi | Intrinsic and light-induced nonadiabatic phenomena in the NaI molecule[END_REF][START_REF] Csehi | Attosecond Molecular Dynamics[END_REF] and spectroscopic properties. [START_REF] Szidarovszky | Direct Signatures of Light-Induced Conical Intersections on the Field-Dressed Spectrum of Na 2[END_REF][START_REF] Szidarovszky | Conical Intersections Induced by Quantum Light: Field-Dressed Spectra from the Weak to the Ultrastrong Coupling Regimes[END_REF][START_REF] Szidarovszky | Rovibronic spectra of molecules dressed by light fields[END_REF][START_REF] Pawlak | Robust field-dressed spectra of diatomics in an optical lattice[END_REF] These works were mainly focused on diatomic molecules where the angle between the internuclear axis and the field polarization axis provides the missing degree of freedom for a CI to emerge between two crossing field-dressed PESs. This rotation angle together with the internuclear separation span the two-dimensional light-induced branching space along which degeneracy is lifted to first order around the LICI due to the dressing field.

Investigating light-induced nonadiabatic effects in polyatomic molecules is expected to be more challenging and richer than in diatomics. On the one hand, the existence of several vibrational degrees of freedom enables LICIs to be defined without further involving any rotational coordinate. On the other hand, natural CIs are ubiquitous in polyatomics that their occurrence and accompanying natural nonadiabatic effects are likely to be strongly mixed with the light-induced ones. It is thus desirable to first consider situations where a clear separation and identification of the effects of natural and light-induced conical intersections can be made, so as to clarify and understand the effect solely caused by the LICI. Experimental works [START_REF] Kim | Control of 1,3-Cyclohexadiene Photoisomerization Using Light-Induced Conical Intersections[END_REF][START_REF] Corrales | Control of ultrafast molecular photodissociation by laser-field-induced potentials[END_REF] have already invoked the concept of a LICI to provide qualitative interpretation to the results of laser-induced isomerization and photodissociation in polyatomic molecules. A general theory on LICIs discusses strategies of its exploitation and application in polyatomic systems. [START_REF] Demekhin | Light-induced conical intersections in polyatomic molecules: General theory, strategies of exploitation, and application[END_REF] It is well established in the field of "natural" nonadiabatic molecular phenomena that the breakdown of the BO approximation induces strong mixing between the vibrational levels within coupled adiabatic PESs. The Franck-Condon approximation is no longer valid, which typically leads at low energies to the so-called "intensity borrowing" effect. [START_REF] Köppel | Multimode Molecular Dynamics Beyond the Born-Oppenheimer Approximation[END_REF] This is a characteristic fingerprint of nonadiabatic effects in molecular spectroscopy, manifested by irregular variations of spectral peak intensities and the appearance of unexpected levels.

Along this line, our current work presents strong intensity-borrowing effects, now appearing in the field-dressed low-energy vibronic spectrum of the H 2 CO (formaldehyde) molecule and absent in the absence of the field, providing direct proof of strong light-induced nonadiabatic effects. The photochemistry of H 2 CO has been studied in various contexts and the corresponding literature is abundant and beyond the scope of the present work. We chose H 2 CO for two reasons: first, because there is no natural CI in the vicinity of the Franck-Condon region. A seam of CIs has been characterized for example by Araujo et al. [START_REF] Araujo | The Photochemistry of Formaldehyde: Internal Conversion and Molecular Dissociation in a Single Step?[END_REF][START_REF] Araújo | The molecular dissociation of formaldehyde at medium photoexcitation energies: A quantum chemistry and direct quantum dynamics study[END_REF][START_REF] Araújo | Controlling Product Selection in the Photodissociation of Formaldehyde: Direct Quantum Dynamics from the S 1 Barrier[END_REF] but it is protected by a transition barrier at low energies; second, this system presents the great advantage of not having any first-order nonadiabatic coupling between the ground and first singlet excited electronic states around its equilibrium geometry. As a consequence, such a situation allows unambiguous identification of light-induced nonadiabatic phenomena, clearly separated from other "natural" nonadiabatic effects.

By using a two-step protocol we simulate the weak-field absorption and stimulated emission spectra of the field-dressed H 2 CO molecule at low energy (infrared domain) taking into account all six vibrational degrees of freedom explicitly. First, we compute the field-dressed states which are superpositions of field-free molecular eigenstates coupled by a mediumintensity laser dressing pump pulse switched on adiabatically. Second, we assume that the low-energy spectrum of the field-dressed molecule can be accessed with a weak-intensity steady-state probe. The dipole transition amplitudes between the field-dressed states are thus evaluated within the typical framework of first-order time-dependent perturbation theory, revealing intensity borrowing in the infrared domain for the field-dressed molecule.

Let us start with presenting the field-free vibrational spectrum of H 2 CO (see the Sup-porting Information for the structure and normal modes of H 2 CO) in its electronic ground state X, shown in the upper panel of Figure 1. As expected, the field-free vibrational spectrum consists of a moderate number of peaks, corresponding to vibrational transitions from the initially populated vibrational ground state to vibrationally-excited eigenstates. Besides the six fundamental transitions (denoted with the normal mode labels ν i , described in Table S1 of the Supporting Information) two combination transitions (ν 2 + ν 6 and ν 3 + ν 6 ) and an overtone transition (2ν 2 ) carry appreciable intensity. It is clearly visible in Figure 1 that no peaks appear below 1100 cm -1 in the field-free vibrational spectrum. However, if the molecule is dressed with a laser field switched on adiabatically and then the spectrum of the field-dressed molecule is measured with a weak probe pulse, striking effects arise in the fielddressed spectrum. The lower panel of Figure 1 shows the spectrum of H 2 CO molecule dressed with a laser field linearly polarized along the body-fixed y axis (defined in the Supporting Information) with the dressing intensity of I d = 10 11 W/cm 2 and dressing wavenumber of ω d = 32932.5 cm -1 (wavelength λ d = 303.65 nm). The field-dressed spectrum in Figure 1, as opposed to the field-free vibrational spectrum, shows the emergence of prominent peaks below 1100 cm -1 . At the same time the intensity of the major peaks seen in the field-free spectrum are substantially lowered. They seem to have borrowed their original intensity to the light-induced peaks below 1100 cm -1 .

In order to understand the origin of the differences between the field-free and field-dressed spectra in Figure 1 a brief description of the underlying theory is necessary. Throughout the current work the two singlet electronic states S 0 ( X 1 A 1 ) and S 1 ( Ã 1 A 2 ) of H 2 CO are taken into account, the corresponding six-dimensional PESs are denoted by V X and V A , respectively. We assume that the electronic states X and A are coupled by a time-dependent external electric field E(t) = E 0 cos(ωt). In the static Floquet-state representation [START_REF] Chu | Floquet theory and complex quasivibrational energy formalism for intense field molecular photodissociation[END_REF][START_REF] Chu | Beyond the Floquet theorem: generalized Floquet formalisms and quasienergy methods for atomic and molecular multiphoton processes in intense laser fields[END_REF] the 0 500 1000 1500 2000 2500 3000 3500

Probe wavenumber / cm In sharp contrast to the field-free spectrum, the field-dressed spectrum exhibits a rich appearance of numerous peaks below 1100 cm -1 . Note the substantial lowering of the intensity due to the field of the peaks present in the field-free spectrum (the intensity scale is the same in both panels).

Hamiltonian matrix has the following form

Ĥ =    T 0 0 T    +    V X + n ω W W V A + (n -1) ω    (1)
where T is the vibrational kinetic energy operator and n is the Fourier index that labels the different light-induced potential channels. The kinetic energy operator is fully exact, properly includes the exchange symmetry of the identical nuclei and it is constructed using a general numerical procedure described in refs. 51 and 52. The operator W = -d(R)eE 0 /2 describes the coupling between the electronic states X and A with d(R) being the body-fixed molecular transition dipole moment (TDM) vector depending on the vibrational coordinates R. Moreover, e, E 0 and ω denote the polarization, amplitude and angular frequency of the dressing laser field, respectively. The permanent dipole moments (PDMs) of the X and A electronic states are included in the full calculations, but for simplicity not shown explicitly in eq 1. We used linearly-polarized electric fields throughout this work. The rotational degrees of freedom were omitted from our computational protocol and the orientation of the molecule was fixed with respect to the external electric field. After diagonalizing the potential energy matrix of eq 1 one can obtain the adiabatic potential energy surfaces V lower and V upper . These two PESs can cross each other, giving rise to a LICI, whenever the conditions W = 0 and

V X = V A -ω are simultaneously fulfilled.
The field-dressed eigenfunctions |Φ k (n) and quasienergies k (n) can be obtained by determining the eigenpairs of the Hamiltonian of eq 1. The eigenfunctions |Φ k (n) can be expanded as the linear combination of products of field-free molecular vibronic eigenstates (denoted by |Xi and |Ai for the electronic states X and A, respectively) and the Fourier vectors of the Floquet states, that is

|Φ k (n) = i C (k) Xi |Xi |n + i C (k) Ai |Ai |n -1 . (2)
In eq 2 i labels vibrational eigenstates and |n is the nth Fourier vector of the Floquet state.

The expansion coefficients C (k)

Xi and C

(k)

Ai can be obtained by diagonalizing the light-dressed Hamiltonian of eq 1 after constructing its matrix representation. In this work the field-free molecular eigenstates of H 2 CO were computed with the GENIUSH program package [START_REF] Mátyus | Toward Black-Box-Type Full-and Reduced-Dimensional Variational (Ro)vibrational Computations[END_REF][START_REF] Fábri | Rotating full-and reduced-dimensional quantum chemical models of molecules[END_REF][START_REF] Császár | The Fourth Age of Quantum Chemistry: Molecules in Motion[END_REF] treating all six vibrational degrees of freedom in a numerically exact way. The V X and V A PESs were taken from refs. 54 and 55, respectively. We note that the C 2v point group can be used to label the eigenstates of the field-free molecule and the symmetries of the X and A electronic states are A 1 and A 2 at the Franck-Condon point of C 2v symmetry.

The transition amplitudes between field-dressed states |Φ k (n) and |Φ l (n) within the framework of first-order time-dependent perturbation theory can be obtained as

Φ k (n)| dα |Φ l (n) = ij C (k) * Xi C (l) Xj Xi| dα |Xj + ij C (k) * Ai C (l) Aj Ai| dα |Aj ( 3
)
where dα is the electric dipole moment operator (α = x, y, z). One can notice that the transitions |Φ k (n) → |Φ l (n) are associated with the PDM related to the electronic states The dressing situation corresponding to the field-dressed spectrum in Figure 1 is featured in Figure 2 where one-dimensional cuts of the V X and V A PESs are shown along the ν 2 (C=O stretch) normal mode (normal coordinates other than Q 2 are set to zero). In Figure 2, the system is dressed with photons corresponding to ω = 32932.5 cm -1 and the V A PES is shifted down with the energy of the dressing photon, see eq 1. Consequently, the vibrational ground state of X becomes nearly resonant with multiple close-lying excited vibrational eigenstates of A and the resulting field-dressed states can be described as superpositions of the eigenstates mentioned. Figure 3 provides two-dimensional cuts of the light-induced adiabatic PESs.

The two-dimensional cuts, given as functions of the Q 2 (C=O stretch) and Q 4 (out-of-plane bend) normal coordinates (the remaining four normal coordinates are set to zero), clearly

show that a LICI is formed between the two light-induced adiabatic PESs. Having discussed the underlying theory and dressing mechanism, we can finally move on to the detailed analysis of the field-dressed spectrum in Figure 1. In what follows we assume that the dressing field is switched on adiabatically and the initial field-dressed state |Φ i (n) is chosen as the field-dressed state which gives maximal overlap with the vibrational ground state of the X electronic state. The newly-emerging peaks, readily visible in the lower panel of Figure 1, can be classified into four main groups. The first two groups consist of peaks between 400 -650 cm -1 (g 1 ) and 700 -800 cm -1 (g 2 ), while the third and fourth groups involve peaks located in the intervals 800 -900 cm -1 (g 3 ) and 1000 -1100 cm -1 (g 4 ). The transitions of g 1 and g 2 types are all polarized along the body-fixed x axis, while the g 3 and g 4 transitions are polarized along the body-fixed y (hardly discernible in Figure 1) and z axes, respectively.

The peaks appearing below 1100 cm -1 in the field-dressed spectrum can be attributed to admixtures of the vibrational eigenstates of the A electronic state in the initial field-dressed state |Φ i (n) . |Φ i (n) in this particular case is a superposition of the X vibrational ground state (0.0 cm -1 , A 1 symmetry, 52.3%) and two close-lying A vibrational eigenstates of B 1 symmetry, namely 32931.2 cm -1 (36.0%) and 32937.2 cm -1 (11.5%) where the numbers in the parentheses indicate the populations of the vibrational eigenstates in |Φ i (n) . Note that the inset in Figure 2 shows the two close-lying A vibrational eigenstates mentioned (two lowest blue lines on the right) that are coupled to the X vibrational ground state (red line on the left) by the dressing field. As the dressing field is polarized along the body-fixed y axis and the body-fixed y component of the TDM transforms according to the B 1 irreducible representation (see Table S3 in the Supporting Information), the X vibrational ground state (A 1 symmetry) can be coupled only to vibrational eigenstates of B 1 symmetry of the A electronic state by the dressing field.

A detailed analysis of the field-dressed states has revealed that the final field-dressed state |Φ f (n) for each peak below 1100 cm -1 remains an eigenstate of the field-free molecule and has the form |Φ f (n) = |Aj |n -1 to a good approximation. Therefore, the peaks below 1100 cm -1 can be interpreted as transitions from the vibrational eigenstates of A at 32931.2 cm -1 and 32937.2 cm -1 (both of B 1 symmetry) to other A vibrational eigenstates making up the final field-dressed states and they are associated with the PDM of the A electronic state according to the second term in eq 3. The richness of the field-dressed spectrum can be primarily attributed to the ν 4 mode as the anharmonic double-well nature of the V A PES along the ν 4 mode can lead to multiple changes in the vibrational quantum number v 4 (|∆v 4 | = 0, 1, 2, 3, . . . ). The emergence of peaks in the interval 0 -1100 cm -1 can be interpreted as intensity borrowing whereby the intensity is borrowed from peaks that are also present in the field-free vibrational spectrum.

Similar intensity borrowing effects are observed for many other laser parameters. Figure 4, for instance, compares the field-free and field-dressed spectra for a different laser frequency, ω d = 34189.5 cm -1 (λ d = 292.49 nm). Similarly to Figure 1, several peaks below 1100 cm -1 appear in the field-dressed spectrum, which can again be understood by the aforementioned mechanism.

In the present work we have outlined a pump-probe scheme to obtain the field-dressed low-energy vibronic spectra of polyatomic molecules treating all vibrational degrees of freedom. The H 2 CO molecule which does not exhibit a natural conical intersection in the studied energy region was chosen as a showcase example. In order to calculate the weak-field ab-

  Although ref. 45 provides a detailed discussion of the possible LICI strategies in polyatomics, no actual results which are based on accurate multidimensional quantum calculations and clearly demonstrate light-induced nonadiabatic effects have been published since then. The present study goes beyond previous investigations and makes an attempt to identify the impact of non-adiabaticity of LICIs in polyatomic molecules by employing accurate multidimensional quantum-dynamical methods instead of simple model calculations.

1 Figure 1 :

 11 Figure 1: Impact of light-induced non-adiabaticity on the vibrational spectrum of H 2 CO.Upper panel: field-free vibrational spectrum of the H 2 CO molecule in its electronic ground state (X) with peaks corresponding to transitions from the vibrational ground state to vibrationally-excited eigenstates as indicated by the normal mode labels. Lower panel: spectrum of H 2 CO dressed with a laser field linearly polarized along the body-fixed y axis with I d = 10 11 W/cm 2 and ω d = 32932.5 cm -1 . Transitions polarized along the x, y and z axes are shown by the markers •, and , respectively. In sharp contrast to the field-free spectrum, the field-dressed spectrum exhibits a rich appearance of numerous peaks below 1100 cm -1 . Note the substantial lowering of the intensity due to the field of the peaks present in the field-free spectrum (the intensity scale is the same in both panels).

XFigure 1 .

 1 Figure 1. For further discussion of the spectroscopy of field-dressed molecules we refer to ref. 41.

50 Figure 2 :

 502 Figure2: One-dimensional field-free potential energy cuts along the ν 2 (C=O stretch) normal mode. The excited-state potential curve (V S1 in the figure) is shifted down by the photon energy value corresponding to 32932.5 cm -1 (see eq 1). Vibrational energy levels are indicated by horizontal lines with colors encoding irreducible representations of the C 2v point group. Due to the high density of S 1 states in six degrees of freedom, levels of S 1 become quasi-degenerate with levels of S 0 and interact with them non-adiabatically.

Figure 3 :

 3 Figure 3: Two-dimensional light-induced adiabatic potential energy surfaces along the ν 2 (C=O stretch) and ν 4 (out-of-plane bend) normal modes. The dressing wavenumber and intensity are chosen as ω d = 29000 cm -1 and I d = 10 14 W/cm 2 , respectively, and the dressing field is polarized along the body-fixed y axis. These dressing wavenumber and intensity values are chosen for a better visualization only. The light-induced conical intersection is highlighted in the inset on the right-hand side of the figure. The character of the adiabatic potential energy surfaces is indicated by different colors (see the legend on the left).

1 Figure 4 :

 14 Figure4: Impact of light-induced non-adiabaticity on the vibrational spectrum of H 2 CO for a somewhat higher laser frequency than that used in Figure1. Upper panel: field-free vibrational spectrum. Lower panel: spectrum in the laser field. The dressing frequency is ω d = 34189.5 cm -1 . All other details are as in Figure1. The figure shows that the prominent appearance of low-lying light-induced levels is not unique, but the details depend sensitively on the laser parameters.

Acknowledgement

Professor Joel Bowman is gratefully acknowledged for providing Fortran subroutines for the S 0 and S 1 potential energy surfaces. This research was supported by the EU-funded Hungarian grant EFOP-3.6.2-16-2017-00005. The authors are grateful to NKFIH for financial support (grants No. K128396 and PD124699).

Supporting Information Available

Equilibrium structure and normal modes of the H 2 CO molecule, technical details of the computations. This material is available free of charge via the Internet at http://pubs. acs.org/.


Marta Avalos-Fernandez
email: marta.avalos-fernandez@u-bordeaux.fr

Marcela Henríquez-Henríquez

A decision-making tool to fine-tune abnormal levels in the complete blood count tests

Keywords: Interpretability, Lasso, GAM, Imbalance, Population Health

The complete blood count (CBC) performed by automated hematology analyzers is one of the most ordered laboratory tests. It is a first-line tool for assessing a patient's general health status, or diagnosing and monitoring disease progression. When the analysis does not fit an expected setting, technologists manually review a blood smear using a microscope. The International Consensus Group for Hematology Review published in 2005 a set of criteria for reviewing CBCs. Commonly, adjustments are locally needed to account for laboratory resources and populations characteristics. Our objective is to provide a decision support tool to identify which CBC variables are associated with higher risks of abnormal smear and at which cutoff values. We propose a cost-sensitive Lasso-penalized additive logistic regression combined with stability selection. Using simulated and real CBC data, we demonstrate that our tool correctly identify the true cutoff values, provided that there is enough available data in their neighbourhood.

Introduction

The complete blood count (CBC) with leukocyte differential count performed by automated hematology analyzers is one of the most ordered laboratory tests since used as first-line tool for assessing a patients general health status, diagnosing and monitoring disease progression and therapy. An automated hematology analyzer is a machine that rapidly counts and discriminates blood cells depending on its size, complexity or staining. When the analysis does not fit an expected setting, the machine triggers a warning flag. Then, technologists manually prepare and review a blood smear using a microscope. The manual technique is more efficient for the identification of cytological nuances, particularly in the presence of immature or abnormal cells. However, this is one of the most time-consuming hematology laboratory tests and requires a high degree of technical skill to minimize errors inherent in the subjectivity of the procedure. There is great interest in reducing the number of automated CBCs requiring manual blood smear reviews without sacrificing the quality of patient care [START_REF] Clé | Blood film in the era of streaming cells[END_REF].

The International Consensus Group for Hematology Review published in 2005 a set of rules as criteria for reviewing CBCs [START_REF] Barnes | The international consensus group for hematology review: sug-gested criteria for action following automated CBC and WBC differential analysis[END_REF]. However, laboratories must adjust these general rules to their patients and machine characteristics. Commonly, adjustments are done accounting for the technical capabilities of hematology automatons, the population/sample distributions of the CBC data and permanently updated expert guidelines. The goal is to achieve the best trade-off between optimizing human and financial resources (i.e. minimizing the false positive rate) and optimizing the quality of patient care (i.e. minimizing the false negative rate, usually set in advance).

The manual smear is also considered as an internal quality control procedure for the evaluation of the automated hematology machine parameters. Regularly, all (flagged and unflagged) samples produced during a limited period of time are analyzed to determine the need for adjustments of the screening criteria for manual blood smear review. These large amounts of data regularly produced could be used to train machine learning algorithms that could be integrated to the quality control procedure as a decisionmaking tool to fine-tune abnormal levels in the CBC tests.

The objective of the present work was to provide a machine learning tool for decision support in fine-tuning abnormal levels in the CBC tests at laboratory-level. Explicitly, we aim to identify which CBC variables are associated with a higher risk of abnormal manual smear and at which cutoff values. This involves that the machine learning tool must provide interpretable results. However, that should surely not be an argument for sacrifying predictive performance.

Our proposal is based on cost-sensitive Lasso-penalized additive logistic regression. Additive functions are considered to belong to the space of piecewise constant func-tions. The natural sparsity encouraged by the Lasso penalty is combined with a stability selection procedure to enforce model stability [START_REF] Meinshausen | Stability selection[END_REF]. The rationale behind this choice is the following. Manual smear results are the binary outcome (abnormal/normal). CBC tests data are the quantitative predictors. The logistic regression model is a popular choice in the biomedical field. Categorisation of continuous variables is common in clinical studies since results are easier to interpret. Thus, we apply additive models fitted by piecewise constant polynomials. Once the piecewise constant basis functions have been set, we have gone through in a linear space setting. Now, a key question is how to determine the optimal number and location of cutoff points. Standard variable selection procedures are applied when only one or a couple of quantitative predictors are present [START_REF] Liquet | CPMCGLM: an R package for p-value adjustment when looking for an optimal transformation of a single explanatory variable in generalized linear models[END_REF][START_REF] Barrio | Use of generalised additive models to categorise continuous variables in clinical prediction[END_REF]. Since the CBC data consist in some twenty biological measures, multiplied by the number of basis functions, the Lasso appears as a valuable alternative (Tibshirani, 1996). To encourage stable results we used Stability selection: only variables frequently selected by the Lasso analysis over data subsamples are retained [START_REF] Meinshausen | Stability selection[END_REF].

Finally, normal manual smear blood tests are much more usual than abnormal tests (less than 10% in the present study). We are faced with an imbalanced learning problem. The two main families of methods in imbalanced binary classification are sampling and cost-sensitive methods [START_REF] He | Learning from imbalanced data[END_REF]. A simple cost-sensitive strategy consists in weighting individuals' contributions to the likelihood function to account for the degree of disequilibrium or importance [START_REF] J P Dmochowski | Maximum likelihood in costsensitive learning: Model specification, approximations, and upper bounds[END_REF].

Method

Let {(X i1 , . . . , X ip , Y i )} n i=1 be an i.i.d. nsample and {(x i1 , . . . , x ip , y i )} n i=1 a realization of the sample. Y is binary (0 for normal, 1 for abnormal smears). Let us note x j = (x 1j , . . . , x nj ) . Consider the additive logistic model: logit(P (Y i = 1|x i1 , . . . , x ip )) = β 0 + p j=1 f j (x ij ), where f j : R → R are unknown centred functions and β 0 the intercept. Let {χ k j (x j )} p j k=1 be a fixed basis of functions and denote χ k j = χ k j (x j ). Any f j can be expanded in terms of these basis and the unknown parameters

{β k j } p j k=1 : f j (x j ) = p j k=1 β k j χ k j = χ j β j . Consider now par- ticular case of piecewise constants χ k ij = 1 if x ij > q k
j and 0 otherwise, with q k j the kth value of a collection of K j fixed values. Considering the negative log-likelihood loss function, we can write the weighted penalized optimization problem as:

max β n i=1 ω i ln e y i p j=1 χ ij β j 1 + e p j=1 χ ij β j -λ β 1 ,
(1) where the weights ω i = ω > 1 if y i = 1 and 1 if y i = 0 are used to account for the degree of imbalance in the minority class. λ > 0 controls regularization. The intercept is omitted. The selection probability is computed and used as a continuous measure of the stability associated to the Lasso estimates β k j . Amato et al. ( 2016) conducted a comprehensive review of parsimonious additive models and reformulated the estimation problem in terms of group Lasso. Sparse group Lasso and overlapping group Lasso are alternative reformulations [START_REF] Chouldechova | Generalized additive model selection[END_REF][START_REF] Lou | Sparse partially linear additive models[END_REF]. Our proposal is close to these approaches. Nevertheless, by choosing piecewise constant functions, which conveniently categorises CBC data, interpretation, estimation, and computational issues are dramatically simplified. The fused Lasso has also been adapted to multiple change-point detection by piecewise constant functions (Petersen and Witten, 2019). However, this approach is not well suited for large data.

Evaluation

CBCs and blood smear reviews for 9 594 patients performed at the clinical laboratory of the Pontificia Universidad Catolica de Chile in 2016 were available. CBC data consisted in hemoglobin (Hg in g/dL), hematocrit (Ht in %), mean corpuscular hemoglobin concentration (MCHC in g/dL), mean corpuscular volume (MCV in f L), erythrocytes (Er in 10 6 /µL), platelets (P in 10 3 /µL), red blood cell distribution width (RDW-CV in %), leukocytes (Le in 10 3 /µL), immature granulocytes (IG in %) and the leukocyte differential count which includes neutrophils (N), basophils (B), eosinophils (Eo), monocytes (M) and lymphocytes (Ly) (in 10 3 /µL and %). Alarms of suspected alterations of blood cells (binary), sex (binary) and age (in years) were also reported. The response, normal/abnormal smear, is imbalanced: only 7% of the smears were abnormal.

Real data served as the core of a simulation study to evaluate if our procedure was able to detect relevant predictors and relevant cutoff values. The features were defined by the original data. Y was generated assuming the logistic model. We considered 4 scenarios: A and B, with only 2 (low-correlated) relevant predictors and C and D with 5 relevant predictors which shared moderate to high correlation between them and low correlation with all the other variables. Only one β k j per variable was different to 0. In scenarios A and D, they were placed in the extreme percentiles, where observations are scarce. Inversely, in scenarios B and C, they were placed in frequently observed values. β 0 was calibrated to achieve 7% of events. Y was generated 100 times and selection probabilities computed.

Results

In all the 4 scenarios, irrelevant variables were infrequently selected (Figure 1). In scenarios A and D, in which the true features have to be learn from scarce examples, their variability is slightly higher and the target value for the true cut-off values are less often achieved. Indeed, in scenario A, the 2 true values are identified as non zero in 52% and 68% of the situations, respectively. In scenario D, the 5 true values are identified as non zero in few situations. The procedure doesn't present good performance when relevant predictors are correlated to other relevant predictors and the true cut-off values are placed in the extreme percentiles. Inversely, in scenarios B and C, the capability to detect the real cut-off values is excellent. Values in the neighbourhood of the real cut-off values are most frequently selected than other irrelevant values. The impact of this error on the selection of cut-off values is minor.

Conclusion

No multivariate analysis studies, even based on classical statistical methods, have been used to adjust the consensus cut-off values. In general, studies in the literature propose new cut-off values and compare them to the consensus ones using some criteria such as recall. However, they do not describe the process of deducing the new (more efficient) cut-off values. Likely, this process is based on acquired expertise which is not reproducible. Our procedure represents a good trade off between traditional interpretable models (Vellido, 2019) and powerful machine learning methods for clinical laboratories. It is able to detect true cut-off values with high probability, provided that enough data are available in their neighbourhood, while comparable in predictive performance to deep learning (results showed in [START_REF] Avalos | Optimising criteria for manual smear review following automated blood count analysis: A machine learning approach[END_REF]). Our tool has a straightforward practical application from an open source code available on request from the corresponding author (shortly on https://github.com/ mavalosf). Appendix A. Appendix

Figure 1 :

 1 Figure 1: Selection probability for the 4 simulated scenarios. Values are mean over 100 simulation. Black and gray crosses indicate estimated values of relevant and irrelevant predictors, respectively. Circles indicate target values.

Figure 2 :

 2 Figure 2: Heatmap of absolute value correlations of continuous predictors in the real CBC data.

© 2020 M. Avalos-Fernandez, H. Touchais & M. Henríquez-Henríquez.

Acknowledgments

The ideas of this work were born from informal discussions between Marta Avalos and Marcela Henriquez during their research visit in Canberra, Australia (CSIRO's Data61 and ANU John Curtin School of Medical Research, respectively) in 2017. Ideas took shape during the participation of Marta Avalos to the "Big data: the information revolution in biomedical research" symposium organised by Marcela Henriquez and her colleagues from the Pontificia Universidad Catolica de Chile in December 2018.

Marta Avalos kindly thanks Cheng Soon Ong (CSIRO's Data61, Canberra) and Aditya Krishna Menon (formerly, CSIRO's Data61, Canberra; currently, Google, New York) for sharing their knowledge on imbalanced class learning. Marcela Henriquez is indebted to Teresa Quiroga from the Department of Clinical Laboratories, School of Medicine, Pontificia Universidad Catolica de Chile, Santiago, Chile for contributions regarding the CBC data and for advice.

The calculations were carried out on the CURTA cluster of Mesocenter for Intensive Calculation in the Aquitaine French region.

Supplementary figures are presented in the appendix.


Reyès -Pérez

Le Calvé
email: slecalve@unistra.fr

Vapour pressure measurements of hydroxyacetaldehyde and hydroxyacetone in the temperature range (273 to 356) K

Keywords: vapour pressure, Antoine's equation, hydroxyacetone, hydroacetaldehyde, glycolaldehyde, ethylene glycol, enthalpy of vaporization

Vapour pressures of ethylene glycol, hydroxyacetaldehyde and hydroxyacetone were measured by using a standard closed system equipped with a vacuum line, built recently in our laboratory. First, our vapour pressures measurements of ethylene glycol and their excellent agreement with the literature in the studied temperature range, i.e. (308 to 385) K, permitted us to validate our experimental set-up and procedure. The data obtained in the temperature ranges (296 to 356) K and (273 to 304) K were very satisfactorily (Absolute Relative Deviations (ARD) < 2.5 %) fitted according to the Antoine's equation: log10 (P °hydroxyacetaldehyde / Pa) = 12.96 ± 0.82 -(3657 ± 238) / T/K and log10 (P °hydroxyacetone / Pa) = 10.13 ± 0.06 -(2201 ± 153) / T/K. Then, the resulting vapour pressures of both hydroxyacetaldehyde and hydroxyacetone at 293 K were P ° = (3.0 ± 0.2) Pa and P ° = (415 ± 20) Pa for hydroxyacetaldehyde and hydroxyacetone, respectively. The quoted errors correspond to 2 obtained from the least square fit analysis and the estimated systematic relative error of 5 %.

Finally, these experimental expressions of temperature dependences of P ° permitted then to derive the enthalpy of vaporization for both compounds.

Introduction

Hydroxyacetone (HOCH2C(O)CH3) and hydroxyacetaldehyde (HOCH2CHO) are atmospheric pollutants of particular interest. Hydroxyacetone is produced in the atmosphere mainly from the OH-initiated oxidation of isoprene, the main non-methane hydrocarbon in the troposphere [START_REF] Butkovskaya | Mechanism of the OH-Initiated Oxidation of Hydroxyacetone over the Temperature Range 236-298 K[END_REF][START_REF] Carter | Development and Evaluation of a Detailed Mechanism for the Atmospheric Reactions of Isoprene and NOx[END_REF][START_REF] Orlando | Mechanism of the OH-Initiated Oxidation of Methacrolein[END_REF] . Hydroxyacetaldehyde, usually named glycolaldehyde, is also formed in situ in the atmosphere from the oxidation of several volatile organic compounds (VOCs) including ethene, 2-methyl-3-buten-2-ol, and isoprene. [4][START_REF] Niki | Fourier transform infrared study of the kinetics and mechanisms for the chlorine-atom-and hydroxyl-radical-initiated oxidation of glycolaldehyde[END_REF][START_REF] Orlando | Laboratory and Theoretical Study of the Oxy Radicals in the OH-and Cl-Initiated Oxidation of Ethene[END_REF][START_REF] Ferronato | Rate and mechanism of the reactions of OH and Cl with 2-methyl-3-buten-2-ol[END_REF] Once produced in the atmosphere, these two hydroxyl carbonyls compounds have a potentially large atmospheric impact since they lead, in the upper troposphere, to HOX radicals known to increase the oxidizing capacity of the atmosphere and therefore they can influence the ozone budget. [START_REF] Von Kuhlmann | Sensitivities in global scale modeling of isoprene[END_REF] So far, hydroxyacetaldehyde and hydroxyacetone have not been considered to be significant aerosol constituents since it is assumed that these compounds have relatively high vapour pressures. However, recent field observations indicate that they are important atmospheric aerosol constituents. For example, Matsunaga et al., 2005 [START_REF] Matsunaga | Isoprene oxidation products are a significant atmospheric aerosol component[END_REF] provide evidence that hydroxyacetone, hydroxyacetaldehyde (and methylglyoxal) contribute significantly to organic aerosols, (10 to 120) Tg annually. These findings have been confirmed, at least for hydroxyacetaldehyde, by Angove et al. 2006 [START_REF] Angove | The characterization of secondary organic aerosol formed during the photodecomposition of 1,3butadiene in air containing nitric oxide[END_REF] in smog chamber experiment. Note that, in this work, Angove et al. (2006) [START_REF] Angove | The characterization of secondary organic aerosol formed during the photodecomposition of 1,3butadiene in air containing nitric oxide[END_REF] report an estimated value of 5.3 kPa for the equilibrium vapour pressure P° of hydroxyacetaldehyde at 298 K, while Butler et al. (2001) [START_REF] Butler | The millimeter-and submillimeter-wave spectrum of glycolaldehyde (CH2OHCHO)[END_REF] report also an estimated value of 4 Pa at room temperature. These two estimated values differ by more than two orders of magnitude. The situation is even worst for hydroxyacetone for which, to our knowledge, only an estimated upper limit exists i.e. P° < 0.5 kPa at room temperature. [START_REF] Mohacek-Grošev | Vibrational analysis of hydroxyacetone[END_REF] The precise knowledge of the equilibrium vapour pressure of a component is important to predict its partitioning between the gas and particle phases in the atmosphere [START_REF] Junge | Basic considerations about traces constituents in the atmosphere as related to the fate of global pollutants, Fate of pollutants in Air and Water Environments[END_REF] . In addition, the saturation vapour pressure P ° can also be used to estimate Henry's Law constant that represents the partitioning between the gas and the aqueous phases at equilibrium. These partitionings are key physical properties to understand and predict the behaviour of a component in the environment. [START_REF] Rice | s Law Constants for pesticides measured as a function of temperature and salinity[END_REF][START_REF] Suntio | Critical review of Henry's law constants for pesticides[END_REF][START_REF] Staudinger | A critical review of Henry's law constants fir environmental applications[END_REF] The main goal of this work is therefore to provide accurate experimental vapour pressures for both hydroxyacetone and hydroxyacetaldehyde. For this, our experimental set-up based on previous static devices [START_REF] Iannone | Vapor Pressure Measurements in a Closed System[END_REF] was built and validated by vapour pressures measurements of ethylene glycol between (308 and 385) K.

Experimental section

Chemicals

Ethylene glycol (> 99.5 %) and solid hydroxyacetaldehyde dimer (> 99 %) were purchased from Merck and Aldrich, respectively. Hydroxyacetone (> 95 %) was provided by Alfa Aesar. Before use, hydroxyacetaldehyde dimer was melted at temperatures ranging between (80 and 90) °C for more than 2 hrs in order to obtain hydroxyacetaldehyde monomer. [START_REF] Tuazon | A product study of the gas-phase reaction of methyl vinyl ketone with the OH radical in the presence of NOx[END_REF] Hydroxyacetaldehyde monomer was then kept as limpid liquid even when temperature decreased typically down to 293 K.

To eliminate the effect of gaseous impurities, all the compounds were further purified by freeze-pump-thaw cycles using liquid nitrogen/ethanol baths at different temperatures and a vacuum pump (< 0.02 Pa).

Apparatus and Procedure

The experimental setup, shown schematically in Figure 1, is a static vapour-liquid equilibrium apparatus. It consists of a Pyrex flask containing the solid or liquid sample, connected to the vacuum line with a glass/metal connection (Caburn MDC®). It includes also a thermostatic bath, temperature and pressure measurement devices and a vacuum system.

The pressure was measured in the equilibrium cell by a capacitance manometer (Edwards®, 622 Barocel, ranges 0 to 1000 Pa and 0 to 10000 Pa) coupled to a digital indicator (BOC Edwards®) giving uncertainties of (0.1 + 0.15 P/P*) Pa and (1 + 0.15 P/P*) Pa, where P* = 100 Pa, for 0 to 1000 Pa and 0 to 10000 Pa ranges, respectively.

Except for the vacuum system, the entire device was placed in an oven maintained at a temperature slightly higher than that of the oil bath in which the Pyrex flask was immersed, in order to avoid any vapour condensation on the tube walls. Both bath and oven temperatures were measured with thermocouples K coupled to a laboratory thermometer (Allcat Instruments®) having an accuracy of ± 0.25 % of the reading values in °C. In addition, the thermocouples were calibrated by the manufacturer few months before the measurements and were checked by dipping them into an ice bath maintained to 273.15 K.

The vacuum system (not shown) is composed of a combination of turbomolecular/vacuum pump (Edwards® High Vaccum, type EXC120/ type RV8) and a liquid nitrogen trap. Two control valves allow to either isolating the sample from the vacuum system (control valve 1) or isolating the equilibrium cell from the vacuum system, leading to the establishment of an equilibrium between the liquid and gas phases (control valve 2).

Results and discussion

Ethylene glycol

The vapour pressure of ethylene glycol was measured in the temperature range 308 -385 K in order to validate the experimental set-up.

After degassing of ethylene glycol in situ in the pyrex flask, the measurements of the saturation vapour pressure were performed by first establishing a highly stable vacuum in the equilibrium cell. The control valve (2) was then rapidly closed allowing the establishment of equilibrium between the liquid phase and the gas phase while the pressure values in the equilibrium cell were monitored on a computer. Figure 2 illustrates a typical data plot obtained for a given temperature (here at 326.1 K).

As shown in Figure 2 (see solid line), the pressure increased quickly at the beginning of the experiment and after a time scale of about 15 min, it leveled off meaning that the liquid phase reached equilibrium with the gas phase. Nevertheless when this equilibrium was reached, the pressure continued to slowly increase linearly with time due to the presence of leaks (see dash-do-dot line in Figure 2). This leak is specific to equipment used in our system, i.e. vacuum fittings with viton seals (NW references, @Edwards). The pressure reading at a given experimental time was then corrected by subtracting the leak contribution as follows:

Pcorrected = P -v  t ( 1
)
where P is the reading pressure, v is the leak rate and t is the time from the beginning of experiment.

The leak rate v was obtained from the slope of the linear part of the plot of P versus t (see dash-dot-dot line in Figure 2). The plot of the resulting corrected pressures (see solid line in Figure 2) reaches now a horizontal plateau that corresponds to the saturation vapour P° at a given temperature.

Similar experiments were repeated for temperatures ranging between 308 K and 385 K (see Table 1). The relative errors on the vapour pressure measurements correspond to 2 +   where  is obtained from the least square linear fit analysis and 5 % is the estimated systematic relative error. The resulting relative errors varied between (5 and 7) %.

Since the plot of log10 P° versus 1/T showed a linear behaviour (see Figure 3), the results

were fitted according to a simplified Antoine's equation:

) / ( 204 3257 ) 72 . 0 02 . 12 ( ) / ( log 10 K T Pa P  -  = 
where the quoted errors correspond to 2 obtained from the least square linear fit analysis and the estimated systematic relative error of 5%.

The absolute relative deviation (ARD) was defined as follows:

i n i calc P P P n ARD  =         - = 0 0 0 0 1 ( 2
)
where n is the total number of experimental data points and P °calc is the calculated vapour pressure.

According to Eq. 2, the ARD value is less than 2%. It appears therefore that our data can be conveniently fitted according to the simplified Antoine's equation.

The vapour pressure of ethylene glycol derived from the experimental simplified Antoine's equation is P ° = (8.02 ± 0.41) Pa at 293 K.

The enthalpy of vaporization vapH is constant on the investigated temperature range. This later, derived from the slope of log10 P° versus 1/T, is then found to be (62.4 ± 4.0) kJ mol -1 .

Comparison with the literature

The equation derived from the literature [START_REF] Stull | Vapor Pressure of Pure Substances[END_REF] in the same temperature range is: The calculated literature values [START_REF] Stull | Vapor Pressure of Pure Substances[END_REF] In the temperature range (308 to 385) K, the differences between the two set of data are at most 5 %, indicating that our experimental set up is suitable to measure equilibrium vapour pressure with good accuracy.

Similarly, our vaporization enthalpy of (62.4 ± 4.0) kJ mol -1 is consistent with the literature values of 65.6 ± 0.3 kJ mol -1 20 , 67.6 kJ mol -1 21 and 61.9 kJ mol -1 . [START_REF] Pedley | Thermochemical Data of Organic Compounds[END_REF]

Hydroxyacetone and hydroxyacetaldehyde

Once the set-up validated, saturation vapour pressures of hydroxyacetaldehyde and hydroxyacetone were measured in the temperature ranges (296 to 356) K and (273 to 304) K, respectively. The results are listed in Table 2. The relative uncertainties on the measurements varied between (5 and 13) % for hydroxyacetaldehyde and (5 and 7) % for hydroxyacetone. The vapour pressures were fitted according to the Antoine's equation: where the quoted uncertainties correspond to 1 obtained from the least square fit analysis.

Similarly to ethylene glycol, the data of log10 P° versus 1/T can conveniently be fitted, as shown in Figure 4, with a two parameters Antoine's equation: where the quoted errors correspond to 2 obtained from the least square fit analysis and include a estimated systematic relative error of 5 %.

Again, the values of ARD determined according to eq. ( 2), found to be equal to 2.0 % and 2.5 % for hydroxyacetone and hydroxyacetaldehyde, respectively, confirm that the experimental values can be conveniently fitted according to the simplified Antoine's equation. Similarly to ethylene glycol, the enthalpies of vaporization of both compounds were derived from the Clausius-Clapeyron's equation: vapH (hydroxyacetaldehyde) = (70 ± 5) kJ mol -1 vapH (hydroxyacetone) = (42 ± 3) kJ mol -1 .

Comparison with the literature

As noted in the introduction, the literature data concerning the vapour pressures of Similarly, in experiments dedicated to Raman spectroscopy, the vapour pressure of hydroxyacetone was estimated to be less than 500 Pa at room temperature [START_REF] Mohacek-Grošev | Vibrational analysis of hydroxyacetone[END_REF] . This reported upper limit is consistent with our determination of (415 ± 20) Pa at 293 K.

Conclusion

Our static device was tested to measure the vapour pressure of ethylene glycol and the obtained data shown an excellent agreement with those reported in the literature.

We have then measured the vapour pressures of hydroxyacetone and hydroxyacetaldehyde as a function of temperature. These data are the first direct measurements for these two compounds. They will be particularly valuable in laboratory studies where they are often necessary. a) The quoted uncertainties correspond to 2σ level + 5% where σ is the standard deviation of the linear fit of the plot of Log10 Plit vs 1/T 19 . (b) The quoted uncertainties correspond to 2σ level + 5%. (c) The deviation between the experimental and literature values is defined as follows:

  These equations permit then to derive the vapour pressures of both hydroxyacetaldehyde and hydroxyacetone at 293 K, which were P ° = (3.0 ± 0.2) Pa and P ° = (415 ± 20) Pa for hydroxyacetaldehyde and hydroxyacetone, respectively.

  hydroxyacetaldehyde and hydroxyacetone are rather limited. For the determination of the millimeter-and sub-millimeter-wave spectrum of hydroxyacetaldehyde,Butler et al. (2001) estimated that the vapour pressure at room temperature was roughly 4 Pa 11 which is in good agreement with our determination of (3 ± 1) Pa at 293 K. On the contrary, Angove et al.(2006) reported an estimated value of about 5300 Pa at 298 K, a value three orders of magnitude higher than our determination, (4.9 ± 0.3) Pa at 298 K. Besides, in measurements of UV absorption cross-section,Magneron et al. (2005) used partial pressures as high as 66 Pa at 298 K, and 2100 Pa at 333 K[START_REF] Magneron | Photolysis and OH-Initiated Oxidation of Glycolaldehyde under Atmospheric Conditions[END_REF] . If this value obtained at 333 K is roughly consistent with the vapor pressure of 980 Pa found in this work, their value reported at 298 K is approximately 20 times higher that our absolute determination. These discrepancies might be due to the presence of volatile impurities in hydroxyacetaldehyde in their study while they were removed, in the current work, after obtaining of hydroxyacelaldehyde monomers, by freeze-pump-thaw cycles.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :

 1234 Figure 1: Schematic diagram of vapour pressure apparatus.

  are summarized in Table 1 together with our experimental values (P °). Our extrapolated experimental value at 293 K (P °= 8.02 ± 0.41 Pa) is in good agreement with that of Stull et al. (P °lit = 7.45 ± 0.38 Pa) at the same temperature.

Table 1 :

 1 Experimental vapour pressures of ethylene glycol measured in this work and comparison with the literature. 100∆P °/P ° corresponds to the percentage deviation between experimental and literature values.

	T / K	P °lit (a) / Pa	P ° (b) / Pa	100∆P °/P ° (c)
	307.7 (±0.1)	26 (±1)	28 (±1)	0.81
	312.6 (±0.1)	39 (±2)	41 (±2)	1.30
	316.5 (±0.1)	52 (±3)	55(±3)	2.52
	318.8 (±0.1)	62 (±3)	61 (±3)	5.12
	322.1 (±0.1)	79 (±4)	77 (±4)	5.38
	326.1 (±0.1)	124 (±6)	124 (±6)	0
	328.2 (±0. 1)	128 (±6)	137 ± (23)	0.74
	334.5 (±0.2)	192 (±10)	196 (±13)	2.11
	335.2 (±0.2)	201 (±10)	201 (±14)	0.11
	340.2 (±0.2)	282 (±14)	278 (±19)	0.37
	347.9 (±0.2)	464 (±23)	454 (±32)	0.03
	359.2 (±0.2)	923 (±46)	900 (±58)	1.32
	367.9 (±0.2)	1528 (±76)	1483 (±91)	1.98
	373.4 (±0.2)	2086 (±104)	1950 (±131)	1.02
	384.5 (±0.2)	3772 (±189)	3467 (±236)	1.29

Table 2 :

 2 Experimental vapour pressures of both hydroxyacetaldehyde and hydroxyacetone measured in this work. The quoted uncertainties correspond to 2s Level + 5%.

		Hydroxyacetaldehyde			Hydroxyacetone
	T / K	P° / Pa	100∆P o /P o (a)	T / K	P° / Pa	100∆P o /P o (a)
	295.85	3.7 (±0.2)	0.44	273.22	127 (±9)	3.26

Acknowledgement

This work was supported by the French Ministry of Research through the LEFE/CHAT program. This work was also sponsored in part by the region of Alsace and the REseau Alsace de Laboratoires en Ingénierie et Sciences pour l'Environnement (REALISE).