Gas phase modification in ESI-MS can significantly alter the charge state distribution of small peptides and proteins. The preceding paper presented a systematic experimental study on this topic using Substance P and proposed a charge retention/charge depletion mechanism, explaining different gas- and liquid-phase modifications [Thinius et al. 10.1021/jasms.9b00044J. Am. Soc. Mass Spec.2020]. In this work, we aim to support this rational by theoretical investigations on the proton transfer processes from (multiply) charged analytes toward solvent clusters. As model systems we use small (di)­amines as analytes and methanol (MeOH) and acetonitrile (ACN) as gas phase modifiers. The calculations are supported by a set of experiments using (di)­amines, to bridge the gap between the present model system and Substance P used in the preceding study. Upon calculation of the thermochemical stability as well as the proton transfer pathways, we find that both ACN and MeOH form stable adduct clusters at the protonation site. MeOH can form large clusters through a chain of H-bridges, eventually lowering the barriers for proton transfer to an extent that charge transfer from the analyte to the MeOH cluster becomes feasible. ACN, however, cannot form H-bridged structures due to its aprotic nature. Hence, the charge is retained at the original protonation site, i.e., the analyte. The investigation confirms the proposed charge retention/charge depletion model. Thus, adding aprotic solvent vapors to the gas phase of an ESI source more likely yields higher charge states than using protic compounds.

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A widely used approach to compute the action f (A)v of a matrix function f (A) on a vector v is to use a rational approximation r for f and compute r(A)v instead. If r is not computed adaptively as in rational Krylov methods, this is usually done using the partial fraction expansion of r and solving linear systems with matrices A − τ I for the various poles τ of r. Here we investigate an alternative approach for the case that a continued fraction representation for the rational function is known rather than a partial fraction expansion. This is typically the case, for example, for Padé approximations. From the continued fraction, we first construct a matrix pencil from which we then obtain what we call the CF-matrix (continued fraction matrix), a block tridiagonal matrix whose blocks consist of polynomials of A with degree bounded by 1 for many continued fractions. We show that one can evaluate r(A)v by solving a single linear system with the CF-matrix and present a number of first theoretical results as a basis for an analysis of future, specific solution methods for the large linear system. While the CF-matrix approach is of principal interest on its own as a new way to compute f (A)v, it can in particular be beneficial when a partial fraction expansion is not known beforehand and computing its parameters is ill-conditioned. We report some numerical experiments which show that with standard preconditioners we can achieve fast convergence in the iterative solution of the large linear system.

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A common way to approximate F (A)b-the action of a matrix function on a vectoris to use the Arnoldi approximation. Since a new vector needs to be generated and stored in every iteration, one is often forced to rely on restart algorithms which are either not efficient, not stable or only applicable to restricted classes of functions. We present a new representation of the error of the Arnoldi iterates if the function F is given as a Laplace transform. Based on this representation we build an efficient and stable restart algorithm. In doing so we extend earlier work for the class of Stieltjes functions which are special Laplace transforms. We report several numerical experiments including comparisons with the restart method for Stieltjes functions.