Three-dimensional Spline Research Articles

Modeling of Three-Dimensional Ocean Current Based on Ocean Current Big Data for Underwater Vehicles

This paper proposes a real-time and high-resolution current system for underwater vehicle simulation and testing based on global ocean current data. The goal was to address the issue of the existing systems for underwater vehicle simulation, whose tests cannot provide real-time and continuous current velocity data. Thus, a three-dimensional ocean current model (3D-OCM) was built for depths of 0~4000 m via the reconstruction of raw current data, fast-access information retrieval, and three-dimensional interpolation. The three interpolation algorithms’ data smoothness and computational times were contrasted. The three-dimensional spline and bilinear algorithm performed the best, taking about 22 milliseconds to acquire the current information anywhere underwater. The comparative analysis revealed that the constructed current system performed strongly in real time and had good velocity data consistency compared with the current data from the National Marine Data Center (NMDC). Furthermore, the running trajectories of the profiling float without interpolation and with three interpolations were contrasted, where the trajectories were more consistent between the three-dimensional spline and bilinear and the three-dimensional Newton and bilinear interpolations. The system can support various marine phenomena for the underwater vehicle’s hardware-in-the-loop (HIL) simulation and testing, and it is meaningful and valuable for increasing the effectiveness of the underwater vehicle’s research and development.

Геоэкологические методы оценки динамики береговой линии на Черноморском побережье Кавказа

Over the past few decades, the degradetion of coastal marine areas has become an acute problem for the Black Sea coast of the Caucasus. The negetive effects of climete change are more pronounced in the subtropics, where sea levels rise and the frequency and intensity of storms increase. In addition, insufficient efforts to mitigete anthropogenic impact have accelereted the degradetion of bio resources. The work is devoted to the issues of increasing the reliability of coastline forecasting by improving the methods of using GIS technologies. Meterials and methods. Radar topographic mission (SRTM) deta were used to creete a digital elevetion map. In addition, archival remote sensing images of the earth (taken by setellites: Corona, Spot-1, Sentinel 2, Landset 8–9), as well as images from the Google Earth detabase, were digitized. The SAGA geo-informetion system was used to process the images, and three-dimensional spline interpoletion in the Gnuplot software was used for subsequent processing of the obtained elevetion deta. Results. The article presents the results of absolute values of coastline displacements of the Sochi Black Sea region reletive to the basic reference points (1985), as well as the values of the coastline displacement rete in the study area (109 km) for the period from 1985 to 2021. It was found thet coastal erosion of the Black Sea increases nonlinearly, while the maximum value of its average displacements (in the 109 km section) over 36 years was about 24.2 m. The abrasion rete for the entire time period is about 0.67 m/year. The total displacement of reference points of the coastline in the local section no. (8–9) over 36 years of observetions can exceed the average value for the entire 109-meter study area by more than 50 % (1.04 versus 0.67 m/year). This allows us to estimete the degree of development of abrasion processes of the coastline over a period of 36 years, which will allow us to predict future changes in the coastal environment and prevent negetive consequences. Conclusion. The results of the study provide a comprehensive understanding of the evolution of the coastline along the entire length of Greeter Sochi, including the rete of abrasion/accretion processes, the spetial position of segments with different degrees of degradetion of the coastal environment and temporal varietions in the processes under study. The results prove the non-linear neture of coastal processes, demonstreting changes in areas more susceptible to erosion over time and the impact on coastal ecosystems. The study uses remote sensing images and GIS technology to creete a comprehensive coastal zone detabase, with Greeter Sochi as an example. This detabase will contribute to improving the quality of future coastal protection projects, as part of the master plan for the resort city of Sochi, providing a more informed and effective approach to coastal zone management. Resume. As a result, it was found thet coastal erosion in the studied section of the Black Sea coast increases nonlinearly, while the average displacement of the 109 km long coastline over 36 years is 24.2 m, and the abrasion rete for the entire time period is about 0.67 m/year. As economic development expands from Sochi to Lazarevskoye, the load on the coastal environment is expected to increase. Therefore, if appropriete measures are not taken in the identified degradetion zones, these territories may face complete unsuitability for the integreted reproduction of tourist resources.

3D modeling of salt domes according to detailed drilling data in the Precaspian region

To create three-dimensional models of arcomorph structures based on detailed data of drilling without involving genetic considerations and indirect information, the potential fields method is effective, which consists of finding an approximating three-dimensional function, the set of isolevel surfaces of which are identified with stratigraphic boundaries, using the coordinates of the formation intersection points. For approximation, mainly three-dimensional splines expressed in terms of Green’s functions were used. The main difficulties are associated with the presence of structural unconformities, to take into account which the initial data, and the models built on their basis, are subjected to special spatial transformations. To avoid the influence of high-amplitude arcomorphs on the geometry of the layers of the pre-salt complex, the modeling of the latter was carried out separately from the intensely deformed evaporates and overlying layers. Assaying of the method was carried out at one of the sites in the Caspian Sea. On the study area, according to the interpretation of logging data from 249 wells, a three-dimensional model was created, including several arcomorph structures, and covering the stratigraphic interval from the Emsian stage of the Lower Devonian to Quaternary deposits. The results confirmed the presence of arcomorphs with overturned beds. Such structural forms are poorly fixed by 2D–3D seismic survey methods. The proposed method can be applied in geological studies in the search and exploration of hydrocarbons in salt dome areas.

The Design Space of Kirchhoff Rods

The Kirchhoff rod model describes the bending and twisting of slender elastic rods in three dimensions and has been widely studied to enable the prediction of how a rod will deform, given its geometry and boundary conditions. In this work, we study a number of inverse problems with the goal of computing the geometry of a straight rod that will automatically deform to match a curved target shape after attaching its endpoints to a support structure. Our solution lets us finely control the static equilibrium state of a rod by varying the cross-sectional profiles along its length. We also show that the set of physically realizable equilibrium states admits a concise geometric description in terms of linear line complexes, which leads to very efficient computational design algorithms. Implemented in an interactive software tool, they allow us to convert three-dimensional hand-drawn spline curves to elastic rods and give feedback about the feasibility and practicality of a design in real time. We demonstrate the efficacy of our method by designing and manufacturing several physical prototypes with applications to interior design and soft robotics.

The Study of the Lithospheric Magnetic Field over Xinjiang and Tibet Areas Based on Ground, Airborne, and Satellite Data

Combined with the ground, airborne, and CHAMP satellite data, the lithospheric field over Xinjiang and Tibet is modeled through the three-dimensional Surface Spline (3DSS) model, Regional Spherical Harmonic Analysis (RSHA) model, and CHAOS-7.11 model. Then, we compare the results with the original measuring data, NGDC720, LCS-1, and the newest SHA model with the degree to 1000 (SHA1000). Moreover, the error estimation and the geological analysis are carried out, and we investigate the possible correspondence between the lithospheric field and the surface heat flow. The results show that the 3DSS model can better describe the detailed distribution of the lithospheric field after comparing it with other models. Some new features are reflected, particularly in the areas of Southern Xinjiang and Tibet, such as a positive anomaly stripe in the southwest, its neighboring Tashkurgan–Hotan–Cele–Minfeng–Qiemo–Ruoqiang belt, and the middle edge of the Kunlun Mountains. The stripe, in terms of rock composition, has a shallow magnetic field source and is related to magnetic intrusions; the lithospheric field in Tibet is weak. Additionally, when the heat flow distribution is compared to our results, there is a good consistency between a positive stripe of heat flow and a positive stripe of the lithospheric field in southern Tibet. The large heat flow values may be related to the shallow Curie surface, which shows that demagnetization is happening close to the surface. However, more of a ferromagnetic mineral, Titanium magnetite, is found there.

Study of the geomagnetic field’s regional gradients in Chinese continent using three-dimensional surface Spline model

We combined domestic ground-based and satellite magnetic measurements to create a regional three-dimensional surface Spline (3DSS) gradient model of the main geomagnetic field over the Chinese continent. To improve the precision of the model, we considered the data gap between the ground and satellite data. We compared and analyzed the results of the Taylor polynomial, surface Spline, and CHAOS-6 (the CHAMP, Ørsted and SAC-C model of Earth’s magnetic field) gradient models. Results showed that the gradients in the south−north and east−west directions of the four models were consistent. The 3DSS model was able to express not only gradients at different altitudes, but also average gradients inside the research area. The two Spline models were able to capture more information on gradient anomalies than were the fitted models. Strong local anomalies were observed in northern Xinjiang, Beijing, and the junction area between Jiangsu and Zhejiang, and the total intensity <italic>F</italic> decreased whereas the altitude increased. The gradient decreased by 21.69% in the south−north direction and increased by 11.78% in the east−west direction. In addition, the altitude gradient turned from negative to positive while the altitude increased. The Spline model and the two fitted models differed mainly in the field sources they expressed and the modeling theory.

Research on the virtual display of a weft-knitted seamless kneepad based on the free-form deformation model

This paper proposes a virtual display method suitable for a weft-knitted seamless kneepad. The purpose of this study is to realize the virtual display of complete weft-knitted products, and it has the advantage of speed to display the virtual display results online in real-time. The basic loop of the model is controlled by up to eight characteristic points and the yarn path is fitted by a three-dimensional spline curve. First, the characteristic point coordinates of the seamless weft-knitted kneepad in the plane state were calculated, and then the corresponding spatial coordinate transformation matrix was selected according to the corresponding mesh of each loop to calculate the characteristic point coordinates in the wearing state. Finally, the virtual display model of the seamless weft-knitted kneepad was rendered by tubeline geometry. The virtual display of the weft-knitted kneepad is realized by the joint programming of Visual Studio and WebGL. The results show that the structure model can achieve the virtual display of the kneepad with high efficiency. This method has strong real-time performance and practicability.

Visualizing spatial differences in corneal electroretinogram potentials using a three-dimensional surface spline

Objective. The spatial distribution of activity at the retina determines the spatial distribution of electroretinogram potentials at the cornea. Here a three-dimensional surface spline method is evaluated for interpolating corneal potentials between measurement points in multi-electrode electroretinography (meERG) data sets. Approach. 25-channel meERG responses were obtained from rat eyes before and after treatment to create local lesions. A 3rd order surface spline was used to interpolate meERG values resulting in smooth color-coded maps of corneal potentials. Potential maps were normalized using standard score values. Pre- and post-treatment responses were characterized by spatial standard deviation and by difference-from-normal plots. Main results. The spatial standard deviation for eyes with local lesions were significantly higher than for healthy eyes. The 3rd order spline resulted in well-behaved corneal potential maps that maintained low error rate when up to 30% of recording channels were excluded from analysis. Post-normalization, responses could be combined within experimental groups, and individual eyes with lesions were clearly distinguished from the healthy-eye mean response. A 3rd order surface spline is an acceptable means of interpolating meERG potentials to create corneal potential maps. The spatial standard deviation is more sensitive to local dysfunction than absolute amplitudes. Significance. This work demonstrates solutions to key challenges in the recording and analysis of meERG responses: visualization, normalization, channel loss, and identification of abnormal responses. Continued development of the meERG technique is relevant to research and clinical applications, especially where local dysfunction (early progressive disease) or local therapeutic effect (subretinal injection) is of interest.

Structure modeling in three-dimensional simulation of weft-knitted seamless kneepads

This paper proposes a simulation method suitable for weft-knitted seamless kneepads. The purpose of this study is to realize the simulation of complete weft-knitted products, and it has the advantage of speed to display the simulation results online in real-time. The loop of the basic loop model is controlled by up to eight characteristic points and the yarn path is fitted by a three-dimensional spline curve. To string the loop model into the fabric, the coordinate of the characteristic point of the loop at other positions is obtained based on the loop model at the origin by using the translation matrix and the rotation matrix. The simulation of the weft-knitted kneepad is realized by the joint programming of Visual Studio and WebGL. The results show that the structure model can achieve the simulation of the kneepad with high efficiency. This method has strong real-time performance and practicability.

To the calculation of the toroidal shell with a local deepening on the external and on the inner surface

Thin-walled toroidal shells are widely used in the construction During operation, various defects appear on the surface of the shells, in particular, local depressions on the outer and inner surfaces, causing stress concentration in the structure. A three-dimensional spline option of the finite element method was developed to determine the stress-strain state of a toroidal shell with a local deepening on the outer and inner surface. The numerical experiments were carried out. The regularities of the changes in a stress-strain state of the shell with the change in the geometric parameters of the deepening were noted.

Modeling of CHAMP satellite data according to the 3D surface spline model of geomagnetic fields

Surface observations and CHAMP measurement data are employed to develop a three-dimensional surface spline (3DSS) model of mainland China. The magnetic field distribution at the satellite level is then demonstrated using the model obtained. The results of this model are compared and verified by deriving the corresponding two (2DTY) and three-dimensional (3DTY) Taylor polynomial models. Issues such as the removal of disruptive geomagnetic fields, the data gap between the surface and satellite levels, and boundary effects are carefully considered during modeling. We then focus on evaluating the modeling effect of the satellite data. Ten satellite points not involved in the modeling procedure are selected, and the residuals, absolute change rates, and RMSEs of these points are calculated. Results show that the distribution of the magnetic field determined by the 3DSS model is highly consistent with that obtained from the IGRF12 model. Expect for component Y, the absolute change rates of other components are less than 0.5%. Specifically, the RMSE of Y of 3DSS is nearly 60% lower than those of 3DTY and 2DTY; the RMSE of other components of the former are also over 90% lower than those of the latter. This finding implies that the 3DSS model has good performance for modeling satellite data and its results are reliable. Moreover, the modeling effect of 3DTY is better than that of 2DTY.

Anisotropic asymmetric transmission of circularly polarized terahertz waves in a three-dimensional spline assembly.

Asymmetric transmission (AT) for circularly polarized (CP) electromagnetic (e-m) waves in chiral metamaterial (CMM) is a well-known phenomenon. However, most of the CMMs exhibit AT along only one direction. In this work, AT for CP waves with a magnitude of more than 0.5 along three principal directions of a newly made three-dimensional (3D) spline assembly is reported at terahertz frequencies. Surface current analysis is presented to explain the mechanism of AT for CP waves in the proposed 3D assembly.

A Cable-Mapping Algorithm Based on Ground-Penetrating Radar

The demand for mapping the buried cables is increasing dramatically with the rapid expansion of urban area, but there are few specific procedures and approaches to map underground cables without disturbing the normal electricity supply. In this letter, a cable-mapping approach based on the ground-penetrating radar (GPR) is proposed, which consists of two parts. First, the parallel scan lines are established along which the GPR is moved to obtain the B-scan images; the hyperbolic shapes on these obtained images are identified and fitted; and the locations and depths of the detected sample points of the buried cables could be derived from these hyperbolas. Then, due to different terrain and surrounding obstacles, as well as the use of pipe-jacking technology, <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> the underground cables may not be straight. The detected points are interpolated by a three-dimensional spline interpolation algorithm to obtain a smooth 3-D curve with location and depth information. Compared with the 2-D model, the 3-D curve could visualize the direction of buried cables more intuitively. Also, in comparison with other interpolation algorithms, the error caused by the proposed interpolation is minimized on the chosen data sets. Experiments on real-world data sets are conducted, and the obtained results demonstrate the effectiveness of the proposed model. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> Pipe-jacking is a trenchless technology for buried utilities [1].

A deformable multimodal image registration using PET/CT and TRUS for intraoperative focal prostate brachytherapy.

In this paper, a deformable registration method is proposed that enables automatic alignment of preoperative PET/CT to intraoperative ultrasound in order to achieve PET-determined focal prostate brachytherapy. Novel PET imaging agents such as prostate specific membrane antigen (PSMA) enables highly accurate identification of intra/extra-prostatic tumors. Incorporation of PSMA PET into the standard transrectal ultrasound (TRUS)-guided prostate brachytherapy will enable focal therapy, thus minimizing radiation toxicities. Our registration method requires PET/CT and TRUS volume as well as prostate segmentations. These input volumes are first rigidly registered by maximizing spatial overlap between the segmented prostate volumes, followed by the deformable registration. To achieve anatomically accurate deformable registration, we extract anatomical landmarks from both prostate boundary and inside the gland. Landmarks are extracted along the base-apex axes using two approaches: equiangular and equidistance. Three-dimensional thin-plate spline (TPS)-based deformable registration is then performed using the extracted landmarks as control points. Finally, the PET/CT images are deformed to the TRUS space by using the computed TPS transformation. The proposed method was validated on 10 prostate cancer patient datasets in which we registered post-implant CT to end-of-implantation TRUS. We computed target registration errors (TREs) by comparing the implanted seed positions (transformed CT seeds vs. intraoperatively identified TRUS seeds). The average TREs of the proposed method are 1.98±1.22 mm (mean±standard deviation) and 1.97±1.24 mm for equiangular and equidistance landmark extraction methods, respectively, which is better than or comparable to existing state-of-the-art methods while being computationally more efficient with an average computation time less than 40 seconds.

ФОРМУВАННЯ ОРТОГОНАЛЬНИХ СІТОК НА ОСНОВІ ФУНДАМЕНТАЛЬНОГО СПЛАЙНУ

To model smooth curved contours the questions are raised about finding derivatives that can be formed based on user additional information. If such information does not exist, then additional numerical methods are used for calculation. To avoid extra calculations, it is relevant to use fundamental splines which are based only on the point frame information.To work with zero-length curves that allows the formation of orthogonal grids on a plane, additional research is needed in order to save this condition for different values ​​of parameter replacement. The author proposes a method of constructing flat orthogonal grids based on a flat isotropic fundamental spline. The conditions for creating a three-dimensional fundamental spline are given. To form a flat isotropic spline, the condition of isotropic chords is used, which subtend the point frame and forms the statement. It is proposed to use a quasi-conformal parameter change to form an orthogonal grid. The parameter is replaced by a combination of the defined functions f1(u), f2(v) - where u and v are some valid parameters. The functions must be formed on the basis of different parameters. The statement is given regarding the formation of an orthogonal grid based on an isotropic spline with the proposed parameter replacement. To prove the validity of the proposed states, examples of simulated grids are given and the coefficient of the first quadratic form F is calculated, the analysis of which allows us to make certain assumptions. Further studies are related to the construction of minimal surfaces based on the proposed quasi-conformal replacement. Keywords: isotropic curve, fundamental spline, orthogonal grid, isothermal grid, quasiconformal parameter replacement.

Точное решение одной задачи оптимизации, порожденной трехмерным уравнением Лапласа

A one-parameter family of finite-dimensional spaces consisting of special three-dimensional splines of Lagrangian type is defined (the parameter $N$ is related to the dimension of the spline space). The solution of the boundary value problem for the Laplace equation given in a three-dimensional parallelepiped admits a representation in the form of a sum of four summands: a function linear in each of the three variables, and solutions of three particular boundary value problems generated by the original equation. In turn, these problems give rise to three problems of minimizing the functionals of residuals given in the indicated spline spaces. This decomposition allows one to study only one of the three optimization problems (the other two are symmetric in nature). A system of linear algebraic equations is obtained with respect to the coefficients of the optimal spline that gives the smallest discrepancy. It is shown that the system has a unique solution. The numerical solution of the system reduces to the implementation of the sweep method (the stability of this method holds). Numerical experiments show that with increasing $N,$ the minimum of the residual functional tends to zero.

Eqtools: Modular, extensible, open-source, cross-machine Python tools for working with magnetic equilibria

As plasma physics research for fusion energy transitions to an increasing emphasis on cross-machine collaboration and numerical simulation, it becomes increasingly important that portable tools be developed to enable data from diverse sources to be analyzed in a consistent manner. This paper presents eqtools, a modular, extensible, open-source toolkit implemented in the Python programming language for handling magnetic equilibria and associated data from tokamaks. eqtools provides a single interface for working with magnetic equilibrium data, both for handling derived quantities and mapping between coordinate systems, extensible to function with data from different experiments, data formats, and magnetic reconstruction codes, replacing the diverse, non-portable solutions currently in use. Moreover, while the open-source Python programming language offers a number of advantages as a scripting language for research purposes, the lack of basic tokamak-specific functionality has impeded the adoption of the language for regular use. Implementing equilibrium-mapping tools in Python removes a substantial barrier to new development in and porting legacy code into Python. In this paper, we introduce the design of the eqtools package and detail the workflow for usage and expansion to additional devices. The implementation of a novel three-dimensional spline solution (in two spatial dimensions and in time) is also detailed. Finally, verification and benchmarking for accuracy and speed against existing tools are detailed. Wider deployment of these tools will enable efficient sharing of data and software between institutions and machines as well as self-consistent analysis of the shared data. Program summaryProgram title: eqtoolsCatalogue identifier: AFBK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFBK_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU GPL v3No. of lines in distributed program, including test data, etc.: 27204No. of bytes in distributed program, including test data, etc.: 1217844Distribution format: tar.gzProgramming language: Python, C.Computer: PCs.Operating system: Linux, Macintosh OS X, Microsoft Windows.RAM: Several megabytes, depends on resolution of dataClassification: 19.4.External routines: F2PY [1], matplotlib [2], MDSplus [3], NumPy [4], SciPy [5]Nature of problem:Access to results from magnetic equilibrium reconstruction code, conversion between various coordinate systems tied to the magnetic equilibrium.Solution method:Data are stored in an object-oriented data structure with human-readable getter methods. Coordinates are converted using bivariate or trivariate splines.Running time:Coordinate transformations on a 66x66 point spatial grid take between 1 and 5 ms per time slice, depending on the transformation used and how many intermediate results have been stored.

Biomechanical modeling constrained surface-based image registration for prostate MR guided TRUS biopsy.

Adding magnetic resonance (MR)-derived information to standard transrectal ultrasound (TRUS) images for guiding prostate biopsy is of substantial clinical interest. A tumor visible on MR images can be projected on ultrasound (US) by using MR-US registration. A common approach is to use surface-based registration. The authors hypothesize that biomechanical modeling will better control deformation inside the prostate than a regular nonrigid surface-based registration method. The authors developed a novel method by extending a nonrigid surface-based registration algorithm with biomechanical finite element (FE) modeling to better predict internal deformations of the prostate. Data were collected from ten patients and the MR and TRUS images were rigidly registered to anatomically align prostate orientations. The prostate was manually segmented in both images and corresponding surface meshes were generated. Next, a tetrahedral volume mesh was generated from the MR image. Prostate deformations due to the TRUS probe were simulated using the surface displacements as the boundary condition. A three-dimensional thin-plate spline deformation field was calculated by registering the mesh vertices. The target registration errors (TREs) of 35 reference landmarks determined by surface and volume mesh registrations were compared. The median TRE of a surface-based registration with biomechanical regularization was 2.76 (0.81-7.96) mm. This was significantly different than the median TRE of 3.47 (1.05-7.80) mm for regular surface-based registration without biomechanical regularization. Biomechanical FE modeling has the potential to improve the accuracy of multimodal prostate registration when comparing it to a regular nonrigid surface-based registration algorithm and can help to improve the effectiveness of MR guided TRUS biopsy procedures.

직교이방성판의 3차원 자유진동 해석에 관한 연구

This paper presents the three-dimensional stress analysis of orthotropic thick plates using the three-dimensional spline strip method based on the theory of elasticity. The orthotropic plates are made of Aragonite crystal and sitka spruce. To demonstrate the convergence and accuracy of the present method, several examples are solved, and results are compared with those obtained by other exact and numerical methods based on the theory of elasticity. Good convergence and accuracy are obtained. The effects of thickness/width ratio, aspect ratio and boundary conditions on normal stress distributions of Aragonite crystal plates and sitka spruce plates are investigated. Moreover, the difference of weak orthotropic and strong orthotropic properties given to the characteristics of stress distributions are also shown.

Quantum effects and anharmonicity in the H2-Li+-benzene complex: A model for hydrogen storage materials

Quantum and anharmonic effects are investigated in H2-Li(+)-benzene, a model for hydrogen adsorption in metal-organic frameworks and carbon-based materials. Three- and 8-dimensional quantum diffusion Monte Carlo (QDMC) and rigid-body diffusion Monte Carlo (RBDMC) simulations are performed on potential energy surfaces interpolated from electronic structure calculations at the M05-2X/6-31+G(d,p) and M05-2X/6-311+G(2df,p) levels of theory using a three-dimensional spline or a modified Shepard interpolation. These calculations investigate the intermolecular interactions in this system, with three- and 8-dimensional 0 K H2 binding enthalpy estimates, ΔH(bind) (0 K), being 16.5 kJ mol(-1) and 12.4 kJ mol(-1), respectively: 0.1 and 0.6 kJ mol(-1) higher than harmonic values. Zero-point energy effects are 35% of the value of ΔH(bind) (0 K) at M05-2X/6-311+G(2df,p) and cannot be neglected; uncorrected electronic binding energies overestimate ΔHbind (0 K) by at least 6 kJ mol(-1). Harmonic intermolecular binding enthalpies can be corrected by treating the H2 "helicopter" and "ferris wheel" rotations as free and hindered rotations, respectively. These simple corrections yield results within 2% of the 8-dimensional anharmonic calculations. Nuclear ground state probability density histograms obtained from the QDMC and RBDMC simulations indicate the H2 molecule is delocalized above the Li(+)-benzene system at 0 K.