Two-stage leaching of calcium and vanadium from high-calcium steelmaking slag.

Vanadium (V) is a critically important element in many industries. A widely used recovery process is a combination of roasting and leaching. However, roasting is an energy-intensive stage. Generally, basic oxygen furnace (BOF) slag is high in calcium (Ca) but low in V. Ca content and its chemical nature can prevent V leaching. This study presents a potential two-stage leaching process for Ca and V from BOF slag. The method is environmentally friendly using low temperatures and enabling leachate recycling. Furthermore, the utilisation rate of the slag can be higher due to Ca recovery. Ca is first leached using ammonium nitrate and nitric acid solution. The V-containing residue is directed to the second stage, where V is leached using ammonium carbonate ((NH4)2CO3). Ca leaching efficiency was 71% achieved with a low temperature (40°C) and in 20 min. > 99% of the dissolved element was Ca. Increasing the total nitrate concentration increased the leaching efficiency. Reducing the L/S ratio improved selectivity. The solid material was analysed after the leaching stages and a clear change was observed after the Ca-stage. The V leaching efficiency was 50%. > 88% of dissolved element was V (L/S 8, [(NH4)2CO3] 200 g/L, 60°C, and 60 min). Increasing [(NH4)2CO3] and L/S ratio slightly improved the leaching efficiency but decreased selectivity. The studied process implements circular economy principles and has been developed for side streams with low V concentrations. However, further optimisation and developments are required regarding the effectiveness of the process.

Disability Pension Did Not Reduce Opioid Use Among Patients With Failed Back Surgery Syndrome Who Were Trialed and Implanted for Spinal Cord Stimulation.

BACKGROUND: Spinal cord stimulation (SCS) is an effective treatment for failed back surgery syndrome (FBSS). In patients with FBSS, opioids have often been initiated, even before SCS is trialed. OBJECTIVE: We studied the effect of retirement on opioid use in patients with chronic pain after failed back surgery. STUDY DESIGN: A retrospective study design. SETTING: The study was conducted at Kuopio University Hospital. METHODS: The study group consisted of all 230 patients with SCS trialed or implanted for FBSS at Kuopio University Hospital Neurosurgery from January 1, 1996 through December 31, 2014. All purchases of prescribed opioids and their daily defined doses, as well as data on working ability, were obtained from the Social Insurance Institution. Patients were divided into 3 groups: SCS trial only, SCS implanted permanently, and SCS implanted but later explanted. We analyzed the differences in opioid use among these groups 2 years before and 2 years after the start of their disability pension (DP). RESULTS: During the follow-up period, a total of 60 patients received a DP. One year before DP, the majority of patients used opioids (n = 43, 72%), and throughout the one-year follow-up after retirement, the number of users increased slightly (n = 46, 77%). In the permanently implanted SCS group, the number of strong opioid users decreased after retirement. Most patients used a moderate dose (0.1-10.5 morphine milligram equivalent/d). Retirement appeared to interrupt dose escalation in all groups, but doses increased further as the follow-up continued. LIMITATIONS: No structured questionnaires were used in this study. Also, many underlying factors contributing to chronic pain were missing. CONCLUSIONS: DP did not reduce the use of opioids in patients with FBSS. Opioid doses were lower and dose escalation less steep with continuous SCS therapy.

Precipitation of struvite using MgSO4 solution prepared from sidestream dolomite or fly ash.

Struvite (NH4MgPO4∗6H2O) is a slow-release fertilizer produced from phosphorus and nitrogen-containing wastewater in the presence of Mg salts. Commercial Mg salts are the single most significant cost of struvite precipitation. In this study, H2SO4 formed as an industrial sidestream was used to prepare MgSO4 solution from waste dolomite (DOL) and fly ash (FA). MgSO4 solution was then used to precipitate struvite from a synthetic (NH4)2HPO4 solution and from actual industrial process waters. The best results were obtained with real process waters where over 99% of phosphate and about 80% ammonium removals were achieved with both MgSO4 solutions after 30 min of reaction time. A higher molar ratio between Mg and P improved the phosphate removal efficiency, especially with DOL-based MgSO4 solutions; however, it had no practical effect on ammonium removal. The struvite content of precipitates was 75.49% with an FA-based chemical and 60.93% with a DOL-based chemical; other valuable nutrients (Ca, K, S, Fe, Mn, and Cl) were captured in the precipitates. The results indicate that both sidestream-based reagents perform well in struvite precipitation and that the formed precipitates could be used as fertilizers.

Extending the straight leg raise test for improved clinical evaluation of sciatica: validity and diagnostic performance with reference to the magnetic resonance imaging.

BACKGROUND: The straight leg raise test (SLR) is one of the most utilized and studied physical tests in patients with low back pain (LBP) for the detection of lumbar disc herniation (LDH), showing high sensitivity and heterogeneous or low specificity. The high incidence of asymptomatic 'pathologic' findings in the magnetic resonance imaging (MRI) scans may cause verification bias to these results. We studied an extended SLR (ESLR) by adding location-specific structural differentiation movements (hip internal rotation or ankle dorsiflexion) to the traditional SLR for it to better differentiate neural symptoms from musculoskeletal. Previously, the ESLR has shown almost perfect interrater reliability between examiners and ability to detect sciatic patients. In this study, we investigated whether a 'positive' ESLR finding is associated with pathology seen on MRI. METHODS: Forty subjects comprised the study population, 20 in sciatic group and 20 in control group. The ESLR was performed 'blinded' to the subjects. After the ESLR, each subject's lumbar MRI was evaluated. The MRIs were analyzed independently by 2 senior radiologists and a spine specialist clinician. The ESLR and MRI results were cross-tabulated. To obtain the odds ratio (OR) with positive ESLR or SLR results for LDH or nerve root compression (NC), a binary logistic regression analysis with subjects' age, gender, height and weight was performed. ESLR's validity was assessed by combination of interrater agreement and percentage prevalence of both LDH and NC. RESULTS: Of sciatic (ESLR+) patients, 85 % had LDH and 75 % NC in the MRI. Not surprisingly, MRI showed a very high incidence of 'false-positive' findings with the ESLR negative group. The ESLR showed 0.85 sensitivity and 0.45 specificity for LDH and 0.75 sensitivity and 0.50 specificity for NC. A positive result in the ESLR was found to be strongly associated with for both LDH and NC: the OR was 8.0 (p = 0.028) and 5.6 (p = 0.041), respectively. CONCLUSIONS: The ESLR shows high validity in detecting neural symptoms and is strongly associated with pathology seen in the MRI when judged positive. We suggest the use of ESLR in clinical practice as a part of clinical examination, where it may prove to be a valuable tool in detecting patients with sciatic symptoms.

Long-term outcome over a decade after first rib resection for neurogenic thoracic outlet syndrome.

OBJECTIVES: The aim was to determine long-term outcomes over a decade after first rib resection (FRR) in patients with neurogenic thoracic outlet syndrome (NTOS). A secondary aim was to investigate correlation of residual rib stump with long-term symptoms. METHODS: This ambispective cohort included patients who underwent transaxillary FRRs for NTOS between 1998 and 2007. Short-term outcomes at 3-month clinical follow-up were retrospectively collected from medical records. Patients who agreed to participate in the study were invited to a long-term clinical follow-up in 2019. Disabilities of Arm, Shoulder, and Hand Score and Cervical Brachial Symptom Questionnaire were used. A chest X-ray limited to a clavicular projection was taken, and the length of the residual first rib was measured. RESULTS: Twenty patients {mean age 41.8 [standard deviation (SD): 10.3 years]} who underwent 27 FRRs participated in the study. The mean follow-up time was 14.9 (SD: 3.6) years. Excellent or good recovery was noted after 16 (59.3% of operated arms) operations in the short-term follow-up and 22 (81.5%) operations in the long-term follow-up. No reoperations were necessary for residual symptoms. The mean Cervical Brachial Symptom Questionnaire score was 26.7 (SD: 28.2) (maximum 120), and the Disabilities of Arm, Shoulder, and Hand Score was 21.1 (SD: 18.4) (maximum 100) points. Twenty-six patients (96.3%) had a noticeable residual first rib stump. The mean length of the residual first rib was 28.9 (SD: 9.5) mm. More than 30-mm rib stump did not indicate a worse long-term outcome. CONCLUSIONS: This study showed good long-term outcome without symptom recurrence after FRR for NTOS. In most patients, after surgery, quality of life and ability to work improved. Residual rib stump length was not associated with the treatment outcome.

Extending the straight leg raise test for improved clinical evaluation of sciatica: reliability of hip internal rotation or ankle dorsiflexion.

BACKGROUND: The straight leg raise (SLR) is the most commonly applied physical tests on patients with sciatica, but the sensitivity and specificity ratings for disc hernia and neural compression leave areas for improvement. Hip internal rotation tensions the lumbosacral nerve roots and ankle dorsiflexion tensions the sciatic nerve along its course. We added these movements to the SLR (extended SLR = ESLR) as structural differentiators and tested inter-rater reliability in patients with LBP, with and without sciatica. METHODS: Forty subjects were recruited to the study by the study controller (SC), 20 in the sciatic group and in the control group. Two independent examiners (E1&E2) performed the ESLR and did not communicate to the subjects other than needed to determine the outcome of the ESLR. First, SLR was performed traditionally until first responses were evoked. At this hip flexion angle, a location-specific structural differentiation was performed to confirm whether the emerged responses were of neural origin. Cohen's Kappa score (CK) for interrater reliability was calculated for ESLR result in detection of sciatic patients. Also, the examiners' ESLR results were compared to the traditional SLR results. RESULTS: The interrater agreement between Examiner 1 and Examiner 2 for the ESLR was 0.85 (p < 0.001, 95%CI: 0.71-0.99) translating to almost perfect agreement as measured by Cohen's Kappa When the ESLR was compared to the traditional SLR, the overall agreement rate was 75% (30/40). Kappa values between the traditional SLR and the E1's or E2's ESLR results were 0.50 (p < 0.0001; 95%CI 0.27-0.73) and 0.54 (p < 0.0001; 95%CI 0.30-0.77), respectively. CONCLUSIONS: ESLR with the addition of location-specific structural differentiation is a reliable and repeatable tool in discerning neural symptoms from musculoskeletal in patients with radiating low back pain. We recommend adding these movements to the standard SLR with aim of improving diagnostic ability.

Electrical brain activity and facial electromyography responses to irony in dysphoric and non-dysphoric participants.

We studied irony comprehension and emotional reactions to irony in dysphoric and control participants. Electroencephalography (EEG) and facial electromyography (EMG) were measured when spoken conversations were presented with pictures that provided either congruent (non-ironic) or incongruent (ironic) contexts. In a separate session, participants evaluated the congruency and valence of the stimuli. While both groups rated ironic stimuli funnier than non-ironic stimuli, the control group rated all the stimuli funnier than the dysphoric group. N400-like activity, P600, and EMG activity indicating smiling were larger after the ironic stimuli than the non-ironic stimuli for both groups. Further, in the dysphoric group the irony modulation was evident in the electrode cluster over the right hemisphere, while no such difference in lateralization was observed in the control group. The results suggest a depression-related alteration in the P600 response associated to irony comprehension, but no alterations were found in emotional reactivity specifically related to irony.

Preoperative Predictors of Better Long-term Functional Ability and Decreased Pain Following LSS Surgery: A Prospective Observational Study with a 10-year Follow-up Period.

STUDY DESIGN: A prospective observational 10-year follow-up study. OBJECTIVE: This study aimed to examine preoperative predictors for better surgical outcomes in patients with lumbar spinal stenosis (LSS) 10 years after surgery. SUMMARY OF BACKGROUND DATA: LSS is a leading cause of low back surgery in patients older than 65 years. Limited data are available for predictors of long-term surgical outcomes in patients with LSS. METHODS: At the baseline, 102 patients with LSS underwent decompressive surgery, and 72 of the original study sample participated in a 10-year follow-up study. Study patients filled out a questionnaire preoperatively, and follow-up data were collected at 3 months, 6 months, 1 year, 2 years, 5 years, and 10 years postoperatively. Surgical outcomes were evaluated in terms of disability with the Oswestry Disability Index (ODI) and pain with the visual analog scale (VAS). Predictors in the models were nonsmoking status, absence of previous lumbar surgery, self-rated health, regular use of painkillers for symptom alleviation, and BMI. Statistical analyses included longitudinal associations, subgroup analyses, and cross-sectional analyses. RESULTS: Using multivariate analysis, statistically significant predictors for lower ODI and VAS scores at 10 years were nonsmoking status, absence of previous lumbar surgery, better self-rated health, and regular use of painkillers for <12 months. Patients who smoked preoperatively or had previous lumbar surgery experienced more pain and disability at the 10-year follow-up. CONCLUSION: These study results can enhance informed decision-making processes for patients considering surgical treatment for LSS by showing preoperative predictors for surgical outcomes up to 10 years after surgery. Smokers and patients with previous lumbar surgery showed a decline in surgical benefits after 5 years. LEVEL OF EVIDENCE: 3.

Normalization of Spinal Cord Displacement With the Straight Leg Raise and Resolution of Sciatica in Patients With Lumbar Intervertebral Disc Herniation: A 1.5-year Follow-up Study.

STUDY DESIGN: A controlled radiologic follow-up study. OBJECTIVE: The aim of this study was to ascertain whether changes in cord excursion with straight leg raise test (SLR) at 1.5-year follow-up time accompany changes in clinical symptoms. SUMMARY OF BACKGROUND DATA: Lumbar intervertebral disc herniation (LIDH) is known to be a key cause of sciatica. Previously, we found that a significant limitation of neural displacement (66.6%) was evident with the SLR on the symptomatic side of patients with subacute single level posterolateral LIDH. METHODS: Fourteen patients with significant sciatic symptoms due to a subacute single-level posterolateral LIDH were reassessed clinically and radiologically at 1.5 years follow-up with a 1.5T MRI scanner. Displacement of the conus medullaris during the unilateral and bilateral SLR was quantified reliably with a randomized procedure and compared between SLRs and to data from baseline. Multivariate regression models and backward variable selection method were employed to identify variables more strongly associated with a decrease in low back pain (LBP) and radicular symptoms. RESULTS: Compared with previously presented baseline values, the data showed a significant increase in neural sliding in all the quantified maneuvers (P ≤ 0.01), and particularly of 2.52 mm (P ≤ 0.001) with the symptomatic SLR.Increase in neural sliding correlated significantly with decrease of both radicular symptoms (Pearson = -0.719, P ≤ 0.001) and LBP (Pearson = -0.693, P ≤ 0.001). Multivariate regression models and backward variable selection method confirmed the improvement of neural sliding effects (P ≤ 0.004) as the main variable being associated with improvement of self-reported clinical symptoms. CONCLUSION: To our knowledge, these are the first noninvasive data to objectively support the association between increase in magnitude of neural adaptive movement and resolution of both radicular and LBP symptoms in in vivo and structurally intact human subjects. LEVEL OF EVIDENCE: 2.

Reduced Spinal Cord Movement With the Straight Leg Raise Test in Patients With Lumbar Intervertebral Disc Herniation.

STUDY DESIGN: Controlled radiological study. OBJECTIVE: To explore whether impairment of neural excursion during the straight leg raise test occurs in patients with sciatic symptoms secondary to lumbar intervertebral disc herniation (LIDH). SUMMARY OF BACKGROUND DATA: Earlier studies have shown that during the straight leg raise (SLR) test in asymptomatic volunteers tensile forces are consistently transmitted throughout the neural system and the thoracolumbar spinal cord slides distally. METHODS: Fifteen patients with sciatic symptoms due to subacute LIDH were studied with a 1.5 T magnetic resonance scanner. First, a spine specialist diagnosed the LIDH using conventional scanning sequences. Following this subjects were scanned using different scanning sequences for planning and measurement purposes. Displacement of the conus medullaris during the unilateral and bilateral SLR was quantified reliably with a randomized procedure and compared between manoeuvres. RESULTS: The results showed 66.6% less excursion of conus medullaris with SLR performed on the symptomatic side compared with excursions measured with SLR performed on the asymptomatic side (p ≤ 0.001). CONCLUSION: In patients with LIDH, the neural displacement on the symptomatic side is significantly reduced by the compressing IVD herniation. To our knowledge, these are the first data in intact human subjects to support the limitation of neural movements in the vertebral canal with LIDH. LEVEL OF EVIDENCE: 3.

Alkali activation of recovered fuel-biofuel fly ash from fluidised-bed combustion: Stabilisation/solidification of heavy metals.

Recovered fuel-biofuel fly ash from a fluidized bed boiler was alkali-activated and granulated with a sodium-silicate solution in order to immobilise the heavy metals it contains. The effect of blast-furnace slag and metakaolin as co-binders were studied. Leaching standard EN 12457-3 was applied to evaluate the immobilisation potential. The results showed that Ba, Pb and Zn were effectively immobilised. However, there was increased leaching after alkali activation for As, Cu, Mo, Sb and V. The co-binders had minimal or even negative effect on the immobilisation. One exception was found for Cr, in which the slag decreased leaching, and one was found for Cu, in which the slag increased leaching. A sequential leaching procedure was utilized to gain a deeper understanding of the immobilisation mechanism. By using a sequential leaching procedure it is possible fractionate elements into watersoluble, acid-soluble, easily-reduced and oxidisable fractions, yielding a total 'bioavailable' amount that is potentially hazardous for the environment. It was found that the total bioavailable amount was lower following alkali activation for all heavy metals, although the water-soluble fraction was higher for some metals. Evidence from leaching tests suggests the immobilisation mechanism was chemical retention, or trapping inside the alkali activation reaction products, rather than physical retention, adsorption or precipitation as hydroxides.

Eckart frame vibration-rotation Hamiltonians: contravariant metric tensor.

Eckart frame is a unique embedding in the theory of molecular vibrations and rotations. It is defined by the condition that the Coriolis coupling of the reference structure of the molecule is zero for every choice of the shape coordinates. It is far from trivial to set up Eckart kinetic energy operators (KEOs), when the shape of the molecule is described by curvilinear coordinates. In order to obtain the KEO, one needs to set up the corresponding contravariant metric tensor. Here, I derive explicitly the Eckart frame rotational measuring vectors. Their inner products with themselves give the rotational elements, and their inner products with the vibrational measuring vectors (which, in the absence of constraints, are the mass-weighted gradients of the shape coordinates) give the Coriolis elements of the contravariant metric tensor. The vibrational elements are given as the inner products of the vibrational measuring vectors with themselves, and these elements do not depend on the choice of the body-frame. The present approach has the advantage that it does not depend on any particular choice of the shape coordinates, but it can be used in conjunction with all shape coordinates. Furthermore, it does not involve evaluation of covariant metric tensors, chain rules of derivation, or numerical differentiation, and it can be easily modified if there are constraints on the shape of the molecule. Both the planar and non-planar reference structures are accounted for. The present method is particular suitable for numerical work. Its computational implementation is outlined in an example, where I discuss how to evaluate vibration-rotation energies and eigenfunctions of a general N-atomic molecule, the shape of which is described by a set of local polyspherical coordinates.

Constrained molecular vibration-rotation Hamiltonians: contravariant metric tensor.

Here, I present a practical recipe for obtaining contravariant vibration-rotation metric tensors, and thus the kinetic energy operators, when some degrees of freedom are constrained rigidly. An element of the contravariant metric tensor is obtained as a sum of dot products of contravariant measuring vectors, which are obtained from their unconstrained counterparts by adding a frozen mode correction. The present method applies in principle for any choice of shape coordinates and a body-frame for which the contravariant measuring vectors can be evaluated. In contrast to the existing methods, the present method does not involve evaluation of covariant metric tensors, matrix inversions, chain rules of derivation, or numerical differentiation. It is applied in the sequel paper [L. Partanen, J. Pesonen, E. Sjöholm, and L. Halonen, J. Chem. Phys. 139, 144311 (2013)] to study the effects of several different approximations to the kinetic energy operator, when the two large-amplitude OH-torsional motions in H2SO4 are of interest.

A rotamer energy level study of sulfuric acid.

It is a common approach in quantum chemical calculations for polyatomic molecules to rigidly constrain some of the degrees of freedom in order to make the calculations computationally feasible. However, the presence of the rigid constraints also affects the kinetic energy operator resulting in the frozen mode correction, originally derived by Pesonen [J. Chem. Phys. 139, 144310 (2013)]. In this study, we compare the effects of this correction to several different approximations to the kinetic energy operator used in the literature, in the specific case of the rotamer energy levels of sulfuric acid. The two stable conformers of sulfuric acid are connected by the rotations of the O-S-O-H dihedral angles and possess C2 and Cs symmetry in the order of increasing energy. Our results show that of the models tested, the largest differences with the frozen mode corrected values were obtained by simply omitting the passive degrees of freedom. For the lowest 17 excited states, this inappropriate treatment introduces an increase of 9.6 cm(-1) on average, with an increase of 8.7 cm(-1) in the zero-point energies. With our two-dimensional potential energy surface calculated at the CCSD(T)-F12a/VDZ-F12 level, we observe a radical shift in the density of states compared to the harmonic picture, combined with an increase in zero point energy. Thus, we conclude that the quantum mechanical inclusion of the different conformers of sulfuric acid have a significant effect on its vibrational partition function, suggesting that it will also have an impact on the computational values of the thermodynamic properties of any reactions where sulfuric acid plays a role. Finally, we also considered the effect of the anharmonicities for the other vibrational degrees of freedom with a VSCF-calculation at the DF-MP2-F12/VTZ-F12 level of theory but found that the inclusion of the other conformer had the more important effect on the vibrational partition function.

The role of solvent exclusion in the interaction between D124 and the metal site in SOD1: implications for ALS.

Structural changes in the metal site of the copper-zinc superoxide dismutase (SOD1) are involved in the various mechanisms proposed for the pathogenesis of the SOD1-linked familial form of amyotrophic lateral sclerosis (ALS). Elucidating how the metal site of SOD1 can be disrupted by ALS-linked mutations is important for a better understanding of the pathogenesis of the disease and for developing more efficient treatments. Residue D124, a second-sphere ligand of the copper and zinc ions, is known from experimental studies to be essential for the integrity of the metal-site structure. In this work, we used density functional theory calculations and molecular dynamics simulations to elucidate which factors keep D124 attached to the metal site and how structural changes may disrupt the binding between D124 and the metal first-sphere ligands. The calculations show that D124 is kept attached to the metal site in a kinetic trap. The exclusion of solvent molecules by the electrostatic loop of the protein is found to create the binding of D124 to the metal site. The calculations also indicate that changes in the structure of the electrostatic loop of the protein can weaken the D124-metal site interaction, lowering the affinity of the zinc site for the metal. Destabilization of the electrostatic loop of SOD1 has been previously shown to be a common property of ALS-linked variants of the protein, but its role in the pathogenesis of SOD1-linked ALS has not been elucidated.

Polymer dynamics in torsion space.

The large scale motion of proteins, or covalently bonded polymers in general, is governed by the dynamics of the torsion angles, with bond lengths and bond angles kept approximately constant. In the present work, the Lagrangian equations of torsion motion are derived for a general macromolecule. The dynamics is implemented numerically for a test protein, using the velocity Verlet method as the integrator. The results indicate time steps of up to about 30 fs can be used for short time (up to at least 20 ps) simulations, before the dynamics and energy start to differ significantly from results obtained with smaller time steps. For longer time simulations, up to 1000 ps, a time step of 10 fs is relatively safe.

Polymer conformations in internal (polyspherical) coordinates.

The small-amplitude conformational changes in macromolecules can be described by the changes in bond lengths and bond angles. The descriptors of large scale changes are torsions. We present a recursive algorithm, in which a bond vector is explicitly written in terms of these internal, or polyspherical coordinates, in a local frame defined by two other bond vectors and their cross product. Conformations of linear and branched molecules, as well as molecules containing rings can be described in this way. The orientation of the molecule is described by the orientation of a body frame. It is parametrized by the instantaneous rotation angle, and the two angles that parametrize the orientation of the instantaneous rotation axis. The reason not to use more conventional Euler angles is due to the fact that Euler angles are not well-defined in gimbal lock (i.e., when a body axis becomes aligned with its space fixed counter part). The position of the molecule is parametrized by its center of mass. Original and calculated positions are compared for several proteins, containing up to about 100,000 atoms.

Kinetic energy operators in linearized internal coordinates.

It is customary to describe molecular vibrations using as exact kinetic energy operators and as accurate potentials as possible. It has become a standard approach to express Hamiltonians in curvilinear internal displacement coordinates, because they offer a simple and physical picture of vibrational motions, including large amplitude changes in the shape. In the older normal mode model of molecular vibrations, the nuclei are thought to vibrate infinitesimally about the reference configuration, and the shape of the molecule is described using linearized approximations of the true geometrically defined internal displacement coordinates. It is natural to ask how the two approaches are related. In this work, I present a general yet practical way to obtain curvilinear displacement coordinates as closed function of their linearized counterparts, and vice versa. In contrast to the conventional power series approach, the body-frame dependency is explicitly taken into account, and the relations are valid for any value of the coordinates. The present approach also allows one to obtain easily exact kinetic energy operators in linearized shape coordinates.